A stringed musical instrument produces sound through the vibrations of strings fixed at both ends. If the velocity at which waves propagate through each string is known, what information is needed in order to determine the fundamental frequency of each string?

Question

A stringed musical instrument produces sound through the vibrations of strings fixed at both ends.  If the velocity at which waves propagate through each string is known, what information is needed in order to determine the fundamental frequency of each string?

Quizplus · Accepted Answer

11ecba2d_1306_09c7_a3bf_3597cdd23a11_MD0008_00 Standing waves are stationary waves that can be produced within strings, pipes, and other common objects. For example, perturbing a string fixed at both ends simultaneously produces multiple standing waves (numbered n = 1, 2, 3) known as harmonics. The frequency f of the first harmonic (n = 1) is called the fundamental frequency (f₁), the lowest-frequency standing wave that can be produced within an object. The given wave frequency of the first harmonic is related to velocity (v) and to wavelength (λ₁) by the expression 11ecba2d_1306_09c8_a3bf_e1fb425597b9_MD0008_00 A distinguishing feature of standing waves is that standing wave characteristics are closely related to the dimensions of the objects in which they are produced. Accordingly, the wavelength (λ_n) of each harmonic in a string fixed at both ends is proportional to the string length (L): 11ecba2d_1306_09c9_a3bf_7f4a9ea597fd_MD0008_00 Using the above equation, the wavelength of the first harmonic (n = 1) may be expressed as 11ecba2d_1306_09ca_a3bf_711432ea9441_MD0008_00 Substituting this expression for the wavelength of the first harmonic into the equation relating frequency to velocity and wavelength gives 11ecba2d_1306_30db_a3bf_b977ccb4992e_MD0008_00 Therefore, when provided with wave velocity, only the string length would be needed to determine the fundamental frequency. (Choices A & C) The bulk modulus (_EMEM_) refers to the compressibility of a substance. In combination with density (ρ), bulk modulus influences the velocity of sound waves through fluids (eg, the atmosphere). However, bulk modulus and density are both unrelated to standing waves (ie, those observed in strings and pipes). (Choice D) The cross-sectional area of a string is unrelated to the wavelength, frequency, or velocity of standing waves that may be produced within that string. Educational objective: The fundamental frequency of a wave is directly proportional to wave speed and inversely proportional to wavelength. For a string fixed at both ends, the wavelength of each harmonic is proportional to the string length.

A Stringed Musical Instrument Produces Sound Through the Vibrations of Strings