Violin frequencies and harmonics

[toothpaste666]

Homework Statement

A violin has four strings that are 32 cm long and are typically tuned to concert G, D, A, and E (196 Hz, 294 Hz, 440 Hz, and 660 Hz).

A)What is the wavelength of the fundamental mode of oscillation on the A string?
Sketch the waveform.

B)What is the wavelength of the sound in air when the D string is played (at 294 Hz)?
Assume that the velocity of sound in air is 343 m/s.

C)Calculate the frequency of the third harmonic (second overtone) on the D string
and the second harmonic (first overtone) on the A string. You will find that the
harmonics have similar frequencies. What is the beat frequency between the two
harmonics?

Homework Equations

f_n = n(v/2L)
(lamba_n) = 2L/n

The Attempt at a Solution

A) for the fundamental wavelength
lambda = 2L = 2(.32m) = .64 m
the picture is one standing wave over the length of the string

B) v= lambda * f
lambda = v/f = 343/294 = 1.17 m

C) for the D string. to find v we use the fundamental frequency
f = v/2L
294 Hz = v / .64m
v = .64 (294) = 188.16
for the third harmonic
f_3 = 3(v/2L) = 3(188.16/.64) = 882 Hz

for for A string
fundamental
f = v/2L
440 = v / .64
v = .64(440) = 281.6
second harmonic
f_2 = 2(281.6/.64) = 880 Hz

so the beat frequency is +/- 2 Hz

 

[Miles Whitmore]

Your solution looks correct to me. Although for part C you could have found the frequencies of the Nth harmonic easier by just multiplying the fundamental frequency by N.

 

[toothpaste666]

ah yeah you are right. thank you!