Auditor Liability: History, Changes and the Enron Case

Auditor Liability: History, Changes and the Enron Case Audits are carried out to ascertain the validity and reliability of the companys information especially financial information and provide an assessment on the effectiveness of the internal control system. Auditors are responsible to express appropriate audit opinion on the financial statements so as to give a true and fair view of the companys financial position based on the audit evidence they obtained throughout the audit procedures performed. Although auditors need not purposely design audit procedures to detect fraud, they are required to disclose the fraud that they have identified immediately. Recently, audit liability has become an international issue that attracts the concern of practitioners and academicians alike. Auditors liability is the possible legal obligation of an auditor for breach of contract or negligence. Auditors liability can be derived from contract law where auditor is bound by the contract or engagement letter and auditors liability is based on breach of contract; common law where the auditors liability concept is based on auditors negligence, gross negligence or fraud that are developed through court decisions; or statutory law where liability is developed through state statutes or Federal securities laws. An auditor can be held liable for breach of contract, negligence, gross negligence or fraud. Breach of contract will cause the auditor to be liable to their clients while negligence, gross negligence or fraud will lead to the auditors liability towards clients and also third parties. 2.0 The Evolution of Auditors Liability This report will basically discuss on the trend of auditor liability to third parties in United Kingdom (UK) and United States (US) as the liability pressure in these two countries is predominantly intense. The trend of auditor liability to clients will not be discussed in this report as it does not change much. This is because the auditors liability to clients occurs only when there is breach of contract, i.e. when the auditor fails to meet the requirements that were established in the contract or normally in the engagement letter. Prior to 1970s, claims against auditors were relatively rare although the issue of auditors liability to clients and certain third parties has always been there. Nowadays, the number of lawsuits against auditors and public accounting firm have increase in the countries such as US, Canada, UK and Australia. Hence, the auditing profession faces litigation crisis and needs to spend a lot of time and costs defending against the lawsuits. The increase in litigation has brought about some negative effects. For instance, the litigation cost and cost of professional liability insurance of an audit firm has increased which will affect the ability and going concern of the audit firm, and increasing number of capable and experienced accountants leaving the profession which may lead to the issue of lack of accounting professionals in future. Unless action is taken, the future of auditing is under threat; this is clearly not in the publics interest (Ward, G., 1999). As there are many negative effects from the litigation, many researches have been done and proposed that the auditor liabilitys scope should be limited to reduce the litigation risk of an audit firm. The most common suggestions include replacing of joint and several liabilities with proportionate liability, capping of auditor liability to a particular limit and so forth. The scope of auditors liability is expected to be limited in future if the suggestions have been implemented. According to Pacini, C., Hillison, W. Sinason, D. (2000), the trend of the legal liability of auditor to third parties has emerged towards a narrower scope. 2.1 United Kingdom (UK) The issue of auditor liability in UK can be determined at the national level where the decision that binds on all the courts in the country is made at the highest national court. Before 1964, it is relatively difficult for a third party to sue an auditor for negligent misstatements. However, the situation has change in 1964 due the case of Hedley Byrne Co. vs Heller Partners Ltd. (1964). After Hedley Byrne case, there was an apparent trend in the expanding of the scope of duty of auditor to third parties. The case of Hedley Byrne generally provides certain tests that must be satisfied for extending the auditors duty of care to third parties: (1) the auditor must aware that the financial statements are to be used for a particular purpose, (2) a known party was intended to rely on financial statements for that purpose, and (3) there must have been some conduct on the part of the auditor linking him to that party, which indicates the auditors understanding of that partys reliance (Messier, W. F., 2007). This case has brought about the concept of reliance where the auditor will be liable if they had been negligent in conducting audit and the third party had relied on the financial statements audited by them. For auditor to be liable to third parties, it must be reasonable for a third party to place reliance on the auditors report and that the auditors were alert that the person would rely on the report. Next, the case of JEB Fasteners Ltd. vs Marks Bloom Co. (1981) and Twomax Ltd. and Goode vs Dickson, McFarlane Robinson (1982) had emphasize on the concept of foreseeability which seems to lead to the view of unlimited auditor liability. The auditors should be able to anticipate that third party will use the audited financial statement as a guideline in decision making process. Both cases had resulted in the growing number of third parties which have a legal right to sue auditor for negligence. In other word, auditors liability to third parties has increased. According to Pacini, C., Hillison, W. Sinason, D. (2000), the widening area of auditor liability to third parties was virtually reversed by the decision of the House of Lords in the Caparo Industries plc. vs Dickman Others (1990) case. From Caparo case, it was held that three necessary tests must be done in order to determine whether a duty of care by auditor can be imposed. First, the foreseeability of the occurrence of damages due to the misstatement must exist. Secondly, a relationship of proximity must exist between auditor and the suing party. Lastly, it must be just and reasonable to put into effect the duty of care on the one party for the benefit of the other. Auditor will only be liable to the third parties if these three criteria have been fulfilled. Since Caparo case, several cases involving alleged auditors negligence have occurred and the UK courts continued to limit the auditors liability to third party. In a nutshell, the Caparo decision has lessened the scope of auditor liability to third parties for negligent misstatements and cutting back the trend of extending the auditor liability to the unlimited class of investing public. 2.2 United States (US) In US, the issue of auditors liability is determined by the state courts or state legislatures separately where different judicial reasoning are applied in different jurisdictions which results in various rules of law in different states. Four different legal standards have evolved in different states of US to judge the auditor liability to third parties which include privity rule, near privity standard, restatement standard and lastly reasonable foreseeability standard. Different states will adopt different standard which will lead to a different result. Until the mid 1980s, the trend of the auditors liability in US still reflected a tendency toward widen scope of auditor liability to third parties. 1986 is the turning point of auditors liability in the US when Illinois passed an accountant privity statute and since then a trend has emerged toward a more narrow scope of duty to third parties (Pacini, C., Hillison, W. Sinason, D, 2000). 2.3 Summary As a conclusion, the evidence supports a trend towards limiting auditor liability to third-party in both UK and US. 3.0 Current Status of Auditors Liability 3.1 Public Perspective Fraud detection The extend of auditors liability largely depends on the responsible that is borne by a particular auditor. However, the liability pressures enforced to auditor has been increasing in several countries. The current public perspective is that auditors are no longer merely responsible to provide true and fair view on financial statement, but also responsible of detecting fraud on the company operation (Reffett.A,2010). Auditors negligence may as a result cause them to get involved in legal suits. A research has shown that the legal environment in five main countries, which include United Kingdom, Canada, Australia, New Zealand and USA, has examined the legal climate in relative to the auditors liability because third parties in those countries are filling lawsuit against auditors negligent misrepresentation in the financial statement. Therefore, current legal framework is trying to narrow down the liability of auditors to reduce their litigation risk (Reffett.A ,2010). In addition to that, auditor is now currently refusing to accept high risk client audit engagement and increase the cost for professional liability insurance so as to reduce their liability. That is because they need to protect their professions. Otherwise, auditors liability will become gradually heavier as if they will be sued when they failed to detect fraud. Yet, should auditor free from their negligence on the financial statement as their scope of responsibilities are mainly not detecting fraud? (Hassink. H, Meuwissen.R, Bollen.L ,2010). To clarify that, some researches have been done. Research has shown that there is certain level of fraud that can be detected by auditors. Therefore, if auditor fails to detect certain level of fraud, he is said to be liable for negligence. There are a certain standard which has been set to determine auditors liability. Those are the level of evidence for the type of fraud cases to be detected by auditors; determine whether auditors have complied with auditing standards regarding fraud to the impact on various context variables that has been detected by auditors. Another issue raised is that whether or not auditors who are able to collect more fraud evidence are considered to be doing the right things. The answers are still in question and it is really based on judgments from the public (Pacini.C , Hillison.W , Sinason.D ,2000). Because of that, expectation gap somehow exist while carrying out the audit work. 3.2 Lender Perspective Interjecting themselves into decision making of a company Auditor is responsible of expressing opinion on the going concern position of client. If auditor fails to express the opinion, auditor will be liable for negligence. However, in fact, auditors are expressing opinion based on the sufficient information provided by the accountants. However, if auditors express an opinion on the existence of the company for the next twelve months, auditors get involved into decision making process of a company (Pacini.C , Hillison.W , Sinason.D , 2000). These responsibilities should be borne by accountants of that company. That is because the time frame provided by auditor to access the going concern of a company will be too short. Therefore, the opinion provided may not be correct. Yet, as the issue of going concern for a company is important for users, especially lenders and suppliers, they may file lawsuit against auditors due to auditors negligence. 3.3 Legal Perspective Judges have bias in making evaluation on auditors decisions Auditors will often make their professional judgment while making decisions. Yet, sometimes, they may fail to foresee anticipated financial problems or any fraudulent act of the client. However, judges has become bias for evaluating the decision made and liability borne by auditors, especially after the failure of Arthur Anderson in discovering that its client has significantly understated the amount of obsolete inventory (Anderson. J.C, Jennings. M.M, Lowe.D.J, Reckers.M.J , 1997). By law, auditor will be liable for their negligence as if they do not exercise due care while conducting an audit. However, when judges tend to believe that auditors purposely breach its professions and professional requirement, they may make wrong judgments while determining the liability of auditor. To reduce the auditors liability, accounting profession has enacted the audit standards to expand auditors responsibilities to reduce the number of audit failures. However, these efforts will not help a lot as if they are done alone. Most importantly, we have to debias the judges in their decision making process. There are lots of methods that have been used to debias or to reduce the judgment errors. That is because judges play important roles in controlling important element of the trial. For example, there are some specific methods designed for judges to determine the liability borne by auditors. Therefore, a correct judgment can be reached. However, in reality, judges do not really believe in due diligence of auditor because of the increase in number of corporate scandals in recent years. Despite the bias view by judges, UK government has talk on contributory negligence to reduce the liability against negligent auditors. In addition to that, US Private Securities Litigation Reform Act enabled auditors to have negotiation on proportional liability so as to protect auditors from their liability towards shareholders. However, the enforcement of those acts seems to be the main reason behind US audit failures. It is claimed that auditors are given substantial protection. That is because the situation has made the investor has no way to sue parties like law and accounting firm which may get involved in fraudulent activities (Anderson. J.C, Jennings. M.M, Lowe.D.J, Reckers.M.J , 1997). 3.4 Shareholder Perspective Auditor needs to give absolute opinion rather than reasonable opinion on financial statements To reduce the power of manipulation of directors, auditors are indirectly becoming the agent appointed by shareholders to be independent parties of company and directors. Yet, shareholders at the same time doubt the independent of the auditors while they carry out their audit work. That is because they do not know the extent of the auditors honesty and integrity (Sikka.P, Filling.S, Pik Liew, 2009). However, as shareholders are constrained by lots of liability arrangement which can safeguard auditors professions, shareholders hold on to the opinion that auditors hold the responsibility to discover any irregularities in the company and any inconsistency raised. Therefore, auditor holds a responsibility that they have to be able to assess the effectiveness of the internal control inside the corporation, supported with relative substantive procedures within a short audit period. After carrying out all the audit work, it is the responsibility of auditor to reach to an accurate conclusion, especially those concerning financial difficulties and going concern of the clients company. Auditors are liable for shareholders in respect of the truthfulness of the presentation of financial statement. Their liability is going to be heavier because of the increase of complexity of the accounting environment and practices (Sikka.P, Filling.S, Pik Liew ,2009). 3.5 Summary As a whole, it is still a question whether less liability for auditors in certain areas will encourage more accurate financial statement. These are still challenges ahead for the auditor, especially facing legal suits filed by third parties due to misrepresentation of financial statement. 4.0 Relevance of the Topic 4.1 Enron Case Summary of the Case During the 1990s, Enron is a relatively small domestic Texan energy company. Then, Enron become one of the largest US corporations with a selection of international energy trading and utility operations. Enron has a strong profits grew and its market value has reached $70 billion (Accountancy as cited in Unerman ODwyer, 2004). However, concerns relating to Enrons profitability and the major share sales by senior executives have driven a slow but continual decline in share prices in August 2000. In October 2001, US stock market was shocked by the announcement of Enron about the accounting adjustments leading to a significant loss for its third quarter of $618 million and a decrease in its reported net asset value of around $1.2 billion. (BusinessWeek cited in Unerman ODwyer, 2004). At the following weeks, the aggressive earnings management practices by hiding large scale losses and liabilities on off balance sheet was revealed. The senior executives of Enron had created about 3500 off balance sheet partnerships (Special Purpose Entities (SPEs)) (Sloan cited in Unerman ODwyer, 2004). Enron filed for Chapter 11 within 2 months. It has become the largest corporation to enter Chapter 11 bankruptcy with $55 billion estimates of outstanding liabilities. Many official inquiries into Enron where started after the company was collapsed (McLean cited in Unerman ODwyer, 2004). Auditor of Enron Arthur Andersen Arthur Andersen, one of the Big Five global auditing firms, was auditor of Enron. In January 2002 allegation surfaced that, Andersen had shattered most of its working papers with Enron when the Securities and Exchange Commission (SEC) initiated the investigations into accounting practices at Enron (The Business cited in Unerman ODwyer, 2004). Further investigations disclosed that Andersen had played a key role in developing the aggressive earnings management techniques executed by Enron. Andersen was exposed to conflicts of interest because it made more fees in 2000 from selling consulting services to Enron ($27 million) than it did from auditing the Enrons accounts ($25 million) (McLean cited in Unerman ODwyer, 2004). Many clients of Anderson were switched to other big auditing firms because they lost faith in Andersen after these revelations. Andersen was collapsed due to this withdrawal of trust. As a result, there are only Big Four global auditing firms after Enron case. SOX and its impact on Auditors Liability Sarbenes-Oxley Act (SOX) was enacted after the collapse of Enron. A lot of major developments in auditors liability have arisen in US as a result of SOX. The main purpose of SOX is to increase the transparency of financial reporting by improving corporate disclosure and governance practices and to encourage an ethical climate (Toda McCarty as cited in Chung, Farrar, Poonam Thorne, 2010). Auditors liability to third parties has increased because SOX specifies the scope of third parties to whom an auditor owes a duty of care, and requires accounting firm to issue additional financial reports, add disclosure in financial reports or issue new reports about themselves (Chung et al., 2010). Since third parties could rely on this new information which is not previously required when making investment or credit decisions, it increases auditors liability to third parties. In addition, SOX has higher penalties for violations of it as compared to previous legislation. This amendment increases significantly the legal liability of auditors practicing in US. International Reactions after Enron Case In response to SOX, many countries all over the world for example Canada, U.K.., Australia and New Zealand have enacted their legislation. New oversight bodies have been created in U.S. and Canada (Chung et al., 2010). Moreover, the legislation in U.K. and Australia has been modified, and the Institute of Chartered Accountants in New Zealand has issued corporate governance guidelines (Chung et al., 2010). The quick actions taken by countries are to fulfill shareholders demand after numerous failures on trying to increase the auditors liability to third parties. Malaysian Reactions after Enron Case In Malaysia, Malaysian regulators have constructed a more organized and disciplined financial reporting structure to ensure compliance and provide more confidence to the public (MIA Editorial Team, 2002). Hence, Malaysias auditor liability has increased in post-Enron era. Nevertheless, numerous fraud events were discovered in local companies such as Transmile, Megan Media and the Port Klang Free Zone in recent years. Thus, then-Prime Minister Tun Abdullah Ahmad Badawi declared the need to set up an audit oversight mechanism to control and administer the financial reporting of the Public Interest Entity (PIE) in his Budget Speech 2008. In year 2010, Parliament of Malaysia has approved the Securities Commission (Amendment) Act 2010 which gives the permission to set up the Audit Oversight Board (AOB). The AOBs mission is to oversee the auditors of PIE and protect investors interest by promoting confidence in the quality and reliability of their audited financial statement (Gomes, 2010). Moreover, AOB will ensure that Malaysian regulatory framework for auditors are in line with international practices. The establishment of AOB virtually increases auditors liability. 5.0 Own Opinions Regarding the Issue of Auditors Liability 5.1 Impacts of Increased Auditors Liability Article entitled Scope of Auditors Liability, Audit Quality, and Capital Investment mentioned that the accounting profession has been facing increasing number of third-party lawsuits since 1960s. Lawsuits against auditors have resulted in direct financial effects as well as other non-financial effects to the profession and society. Audit firms are experiencing increased costs to settle lawsuits which include the management time and insurance premiums. Moreover, audit firms have suffered negative impact on reputation due to the negative publicity arising from litigation. Also, from the article entitled Auditor liability to third parties: an international focus, it is claimed that the increase in litigation for auditors has led to some detrimental effects. One of the effects is accounting firms now have became more persistent in rejecting the engagement with clients that are viewed to possess high-litigation-risk. Besides that, theres a decrease in the availability and increase in the cost for professional liability insurance. Eventually, many of the experienced accountants gradually depart from the profession. In addition, Bialkin and Cooper (1986) warned that the trend of expanding the scope of auditors liability to third parties will result in a fall in the quality level as well as the scope of services provided by the profession. 5.2 The Need to Minimize Auditors Liability Therefore, theres a need to minimize auditors legal liability. We need to minimize the legal liability of auditors because if the legal liability of auditors continues to rise without any control enforced, the profession might face a severe lack of experienced people in the profession. Accountants and auditors will start quitting the field fearing that they might be the next to face litigation due to negligence. This will eventually results in a chaos in the corporate world when the day where no competent auditors are available to audit the financial statements arrived. In order to minimize auditors legal liability, steps have been taken by the professionals and by individual firms. 5.3 Ways to Minimize Auditors Liability A special committee was formed by the American Institute of Certified Public Accountants in the late 1985 in order to develop a legislative program for liability reform. The program focuses to limit the scope of auditors liability to third parties by retaining the privity standard. Privity approach, Restatement approach, Foreseeability approach Derek K. Chan (2002) concluded in his paper that from an efficiency approach, the privity approach is the best approach to confine an independent auditors work if compared to Restatement approach and foreseeability approach. This is because privity approach provides the firm a credible mechanism where the firm can identify lenders into two groups within its own discretion 1. Lenders who are eligible to recover their losses from the auditors; 2. Lender who are excluded from recovering their damages from the auditors. Restatement approach provides less flexibility for the firm to decide which lenders are eligible, hence expands auditors liability. The foreseeability approach offers no flexibility at all to the firm as this approach broadens auditors liability to include foreseeable third parties who will rely on the audited financial statements. In our opinion, the privity approach is in fact the best approach thus far in order to limit auditors liability. Furthermore, in order to minimize auditors liability, auditors should be setting proper audit plans and perform the best audit strategy. They should follow closely every audit guidelines provided, and never cross over the forbidden lines. 5.4 The Need to Impose Litigations on Auditors Generally, auditors nowadays are quick to issue standard unqualified opinion on financial statements due to the legal protection system. Moreover, although litigations are imposed to auditors, due to the limited liability, auditors are continuing with their tendency to issue the standard unqualified report. Example of legal protection system According to the article entitled Auditors liability-no need to detect fraud, the author concluded that fraudster is not supposed to sue his auditors for failing to detect his fraud. Applying ex turpi causa non oritur actio, if one is engaged in illegal activity, one cannot sue another for damages that arose out of that illegal activity. In the case Moore Stephens (a firm) v. Stone Rolls Ltd. (in liquidation) [2009] UKHL 39, the House of Lords has decided that the illegality defense still applies. In other words, it is determined that Stone Rolls Ltd could not rely on its own illegal acts to claim for damages from its auditor. However, in our opinion, this principle ex turpi causa non oritur actio cannot universally pardon auditors of a duty to detect fraud nor to totally protect them from any litigation. Auditors should undergo a fair trial and be judged reasonably if they were found guilty of conspiracy in the fraud case. Additionally, co-operation between auditors and directors increase the tendency of auditors to issue the standard unqualified opinion. Modifications and adjustments are made for the benefits of management and auditors at the expense of shareholders. As a result, apart from the effort to minimize the auditors liability, in our opinion, theres still a need to impose appropriate legal actions on auditors who are involved in fraud cases because this is the right thing to do in order to maintain the fairness in the corporate world as well as to place emphasis on the importance of ethics and integrity in the profession. 5.5 Conclusion Suggestions Therefore, from our groups perspective, the current condition of auditors liability indicates an imperative need to strike a balance between imposing litigation on auditors who commit fraud or who fail to comply with the approved auditing standards and minimizing auditors legal liability at the same time. In addition, auditors should be well aware of their real masters which are the shareholders. They should get on the job and reveal the fraudsters if any were found. Auditors should be carrying the burden of being accountable for pensioners and investors lost funds. Our group believes that it is time to strike a change in the auditing industry. Auditors should chuck away their old habits and ensure that figures are appropriately presented. Moreover, auditors who had committed fraud should be blacklisted. Audit firms should be brave to accept blames and consequences whenever any fraud is found due to negligence of auditors. Last but not least, we should encourage perfect market competition in this industry. More auditing firms should rise and its time to stop large firms oligopolistic position. 6.0 Conclusion Increased auditors liability poses an advantage for the shareholders and other third parties who use the audited financial statements. This is because with more auditors liability, auditors will tend to be more cautious in their works. They will place emphasis on the importance of conducting appropriate audit procedures and to disclose any discrepancy found. Besides that, they will abide by the law and follow the auditing standards closely in order to avoid any unnecessary litigation. However, as discussed, it is proven that an increase in the auditors liability brought a lot of negative impact on the auditing and accounting profession. People started to response in ways that will protect themselves such as quitting the job or reject high risk audit engagement. Hence, its still an issue as whether less liability or more liability for auditors in certain areas will encourage more accurate and better assurance in the audited financial statements. Why Do the Same Notes Sound Different in Instruments? Why Do the Same Notes Sound Different in Instruments? Why do instruments sound different despite playing t/he same notes? When we listen to music, we are exposed to various different instruments. Choice of instrumentation is one of the main factors that contribute to the overall feeling of a song. Even though major and minor chords and scales can be played across most instruments, the instrument that is chosen to play the note plays a large part in what the musician wants to express. For instance, a double bass playing an E may come across as serious, but a flute playing the same note may come across as cheerful. I aim to investigate why such is the case. Across all instruments, notes are produced by creating standing wave, which will be discussed below. It is these standing waves that cause the air surrounding the object to vibrate, causing a sound wave to spread out. The main factor that causes different sounds in an instrument is the harmonic frequencies and overtones that an instrument outputs on notes, with other factors such as material affecting this factor. A vibrating string does not produce a single frequency, but a mixture of fundamental frequencies and overtones. Say that that an A note is being played on a violin string. If just the fundamental harmonic is heard, it would sound dull. It would also sound similar to other instruments playing the same note in the same pitch, provided only the fundamental frequency is being heard. However, when the string is bowed, multiple harmonic frequencies are produced at the same time. You cannot necessarily hear each harmonic note being played, as all these harmonics blend in to produce the overall sound you are hearing. This image shows the harmonic frequencies, also known as the harmonic scale, that are involved when an A note is played on a violin. These are found by stopping the vibrating string at certain intervals. Pythagoras noticed this when he stopped a vibrating string halfway along its length, which brought the pitch to an octave higher. He did this every half interval of the previous half interval, and found that the pitch consistently became an octave higher. He also found that when the string was stopped a third way through, an octave and a fifth was produced, which also produced increasing pitches in octave intervals. To examine the formula relating the wavelengths, we need to understand how waves are formed on a string. A standing wave is produced when a driver transfers energy to the medium. Energy is transferred down the string, and as it is trapped between two points, reflects of one end and superimposes with waves coming in the opposite direction. However, standing waves6 do not occur at any frequency. Only at specific frequencies do standing waves occur. When transferring energy to the medium at the right frequency, the fundamental frequency is produced. In the case of a violin, the bow that bows the string is the driver, and the string is the medium. Let L be the length of the string, and ÃŽ » be the length of the wave. Let F be the frequency, and velocity of the string be V. As velocity is constant, and v = F ÃŽ », F is inversely proportional to ÃŽ ». At the fundamental frequency5, ÃŽ »0 = (2/1)L and F0 = (v/ ÃŽ »0) , as depicted in this image http://www.chemistry.wustl.edu/~coursedev/Online%20tutorials/Waves.htm As the string is stopped at certain intervals, as Pythagoras did, nodes and antinodes are produced. This is a result of constructive and destructive interference occurring, nodes being points of displacement where destructive interference occurs, due to a Ï€ phase difference, and antinodes being where constructive interference occurs, due to a 2Ï€ phase difference, as shown in this image. http://www.chemistry.wustl.edu/~coursedev/Online%20tutorials/Waves.htm This causes ÃŽ »1 = (2/2) L, F1 = (v/ ÃŽ »1) Considering the fact that the first harmonic causes a relation of ÃŽ »0 =2L, we can see from this image that the wavelength has halved. When the string is stopped a third of the length through, such an image occurs. This produces ÃŽ »2 = (2/3)L, F2 = (v/ ÃŽ »2). The pattern is consistent for the fourth, fifth, sixth and so on harmonic. From here a formula can be formed relating the wavelength of the harmonic to the length of the string. For the first harmonic, ÃŽ »0 = (2/1)L and F0 = (v/ ÃŽ »0). In the second, the denominator of coefficient of L increases by one, whilst F varies according to the wave being produced, in the relationship of Fn ∠ (1/ ÃŽ »n), and the same thing occurs for the third harmonic. So, for the nth harmonic, n being a natural number, this formula shows what the wavelength will be; ÃŽ »n =(2/n)L. As discussed, previously, a note produced on an instrument creates various harmonics. Therefore, by taking the sum of the amplitudes of each harmonic, we can find the shape of the wave produced when a note is played. This is depicted below. 2 A fundamental note of, say, 100 Hz 2 A second harmonic of, say, 200 Hz 2 Adding both waves together produces the resultant patterns above. Just from this very simple example above, we can already visually see how mixtures of harmonics and overtones create interesting waves. From this we can see that different instruments harmonic scales must have different properties. The various harmonics on each instrument do not necessarily have the same strength. For instance, a clarinet is strong in the odd numbered harmonics, but weaker in the even numbered harmonics, whilst a flute is stronger the other way round. 1 As a real life example, this graph shows all the frequencies that are produced when a violinist bows a D note, at 294 Hz. http://www.nagyvaryviolins.com/tonequality.html From close inspection we can see that the first harmonic occurs at around 300Hz. The second harmonic occurs around 600Hz, and the next harmonic at around 900Hz. This fits in with the relationship of Fn ∠ (1/ ÃŽ »n). In each harmonic step, the wavelength decreases, but the frequency increases which is the case above. From the first to second harmonic the wavelength goes from 2L to L, which is a decrease of scale factor  ½. Yet the frequency has increase by a scale factor of 2, which fits the relationship. This graph shows the frequencies produced when a vocalist produces the same note, at the same frequency. http://www.nagyvaryviolins.com/tonequality.html You can see by comparison that there are similarities in the shape of each graph, but with subtle differences. Although the harmonics occur in the same pattern as above, their peaks are slightly different, and at frequencies beyond 5000 Hz the frequencies outputted by the vocalist have a much lower dB than the violin. Therefore we can conclude that the harmonic scales, as shown on the first page, must vary for each instrument. The various harmonics1 on each instrument do not necessarily have the same strength. For instance, a clarinet is strong in the odd numbered harmonics, but weaker in the even numbered harmonics, whilst a flute is stronger the other way round. However, even the same instruments have certain characteristics that distinguish them from other instruments in the same category. For example, a Gibson Les Paul, a type of guitar, produces a much heavier tone than a Fender Stratocaster, and you can even tell the difference between a cheap Les Paul and a custom shop Les Paul if you listen closely. This is down to the materials used to create the instrument, and the dimensions chosen. This is explained below, using the example of a violin. The belly and back plates of a violins4 body are designed to easily resonate. To identify these frequencies, the Authors of this page mechanically drove isolated violin bellies. The force applied is the driving force, and to see the frequency response an accelerometer was used. The acceleration was then monitored, enabling them to plot the ratio of force to acceleration against frequency. Chlandi patterns7 were then used to identify which frequencies the plates resonated at most easily. Chlandi patterns are symmetrical patterns formed when a standing wave is formed on the plate. To see these patterns, granules of sand are placed on the plate, much like iron fillings are used to show the magnetic field of a magnet. The most important frequencies patterns were placed on the graph below. We can see that the resonant frequency is 163Hz as it is forms the most symmetrical pattern, and is the first symmetrical pattern to occur. Resonant frequency is found using the formula6 F = 1/2Ï€ sqrt (k/m), where K is the spring constant and m is the mass. Therefore the closer the frequency is to this frequency, the closer it is to the resonant frequency. Therefore k and m vary according to dimensions and materials used, the resonant frequency varies between instruments, even of the same make. Gibson Les Paul guitars3 are made from mahogany, a dense wood, whilst Fender Stratocaster guitars are made from either ash or alder wood. These types of wood are lighter and less dense than mahogany, which results in the brighter tone Fender Stratocasters are known for, and the heavier and darker tone Gibson Les Pauls are known for. It is the way that the wood responds to the vibrations that travel through it when a note is played that causes this. As discussed above, the body of a violin has certain frequencies it resonates best at. It can be applied here, and can be deduced that the denser material does not pronounce higher frequencies with the same clarity as the less dense material, therefore causing this difference in tone. To conclude, it seems that the main factor that affects the quality of a note produced on an instrument is the harmonics that an instrument produces. The fundamental frequency and harmonics that are expressed the most depends on the dimensions of the instrument and the materials. Even in the cases of the same instrument they can sound different depending on the skill of the maker. It is fascinating to realise that the reason why there are so many instruments in the world and why we are able to experience all these different sounds and feelings are essentially down to the physics of standing waves and resonance. Sources: http://homepage.ntlworld.com/terence.dwyer/The%20Harmonic%20Series%20Explained.pdf http://www.phys.uconn.edu/~gibson/Notes/Section4_2/Sec4_2.htm http://www.differencebetween.net/object/difference-between-gibson-and-fender/ http://newt.phys.unsw.edu.au/jw/violintro.html http://www.physicsclassroom.com/class/sound/Lesson-4/Fundamental-Frequency-and-Harmonics Adams S. and Allday J., Advanced Physics, Oxford, Oxford University press, pg http://skullsinthestars.com/2013/05/02/physics-demonstrations-chladni-patterns/