Question

I know how to do the first 3 parts. Need last 3 parts of the question. Posted whole thing for reference. Thanks!

a) A 1 meter long guitar string of linear mass density 2g/m3 is put under tension until it resonates with a fundamental frequency of 440 Hz. Determine the tension that produces this fundamental frequency. Also determine the other of the first four harmonic frequencies and draw diagrams illustrating what each of these oscillations looks like on the string.

b) This string will produce sound waves in the air, determine the wavelength of the sound waves.

c) Suppose you had two of these strings, placed 25 meters apart. How far from the midpoint between them must you stand to experience the first region of maximum destructive interference? How far would you have to travel to reach the second region of maximum deconstructive interference?

d) If you were to keep walking towards one of the strings, you would experience these regions of maximum destructive and constructive interference as pulses in the amplitude of the sound. If you were to walk at a speed of 1.75 m/s, at what frequency would these pulses occur? This is to say, how many regions of deconstructive interference do you pass per second?

e) To look at the problem another way, if you were to walk directly towards one of the strings and away from the other, both will undergo a Doppler shift. Calculate the Doppler shifted frequency of the string you’re walking towards, ft as well as the frequency of the string you’re walking away from, fa.

f) Given that you’re hearing these two slightly different frequencies at the same time, calculate the beat frequency that you hear from the slight difference between the two frequencies. What do you notice about this “beat frequency” between the Doppler shifted frequencies and the pulsing frequency we observed from passing through the standing wave pattern in part D?

Answer 1

Given that,

length of the string is, L=1 m

frequency is, f=440 HZ

linear mass density, $\lambda$m=m/L=2 gm/m³

a)

The fundamental vibrational mode of a stretched string is such that the wavelength is twice the length of the string.

the expression for the fundamental frequency, f=v_{wave on string}/2L

but wave velocity, V_{wave on string}=$\sqrt{T/(m/L)}$=$\sqrt{T/\lambda m}$

substitute in fundamental frquency f=$\sqrt{T/\lambda m}/2L$

rearrange the terms, tension T=$\lambda m$*(f*2L)²

substitute the values in above expression,

T=(2 gm/m³)*(440 HZ*2*1 m)²

=1548.8 N

The fundamental or first mode has frequency f₁ = v/λ₁ = v/2L,
The second harmonic has frequency f₂ = v/λ₂ = 2v/2L = 2f₁
The third harmonic has frequency f₃ = v/λ₃ = 3v/2L = 3f₁,
The fourth harmonic has frequency f₄ = v/λ₄ = 4v/2L = 4f₁

2

b)

velocity of sound in air is, v=340 m/s

light velocity is, c=3$\times$10⁸ m/s

then, v=c/$\lambda$

rearrange the term,$\lambda$= c/v

                                  =(3$\times$10⁸ m/s)/(340 m/s)

                                = 8.8 A⁰

c) the distructive interference,

$\phi$=$\Delta L/\lambda$*2*pi

=25 m/8.8 A⁰*2*3.14

=4.5

d)

f=v/$\lambda$

=1.75 m/s/0.0088 m

=198.8 HZ

e)

walking towards

f¹=f((v+v_D)/(v+v_S))

walking away

f¹=f((v-v_D)/(v-v_s))

f)

f_beat=f1-f2

      =440 HZ-198.8 HZ

     =241.2 HZ