Vibrations of a Stretched String: Finding Harmonic, Frequency & Wave Speed

[Fusilli_Jerry89]

Homework Statement

A 7 metre long string is stretched between 2 walls so that its ends are fixed. It is made to vibrate and it is found that the displacement, y is given by: y = 0.023sin(xpi)cos(0.714 pi t) where x and y are in metres and t is in milliseconds.

a) to which harmonic, N, does the vibrtion correspond?
b) What is the frequency of the sound emitted by the string?
c) What is the speed of waves in the string?

Homework Equations

y(N) = 2y sin((Npix)/L)cos((vNtpi)/L)

The Attempt at a Solution

To get part a, can u use the formula and see that L must equal N in order to get the equation in the question, or no? I'm lost.

 

[learningphysics]

Yes, you're right. N must equal L to get the equation you have:

y(N) = 2y sin((Npix)/L)cos((vNtpi)/L)

y = 0.023sin(xpi)cos(0.714 pi t)

So N = 7.

Another way to look at it... Find the angle at the end of the standing wave... The angle is xpi... that's coming from 0.023sin(xpi)cos(0.714 pi t)

So what's the angle you get when you plug in x = 7 (end of the standing wave since the walls are 7m apart). you get 7pi. That means the 7th harmonic.

Solve b using "0.714 pi t"

You should be able to solve c, in the same way as a... by comparing: y = 0.023sin(xpi)cos(0.714 pi t) to y(N) = 2y sin((Npix)/L)cos((vNtpi)/L)

what does v need to be...

 

[ogb p]

As a scientist, it is important to first understand the concepts and equations related to the problem. The given equation represents the displacement of the string, which is a function of position (x) and time (t). The sine and cosine functions represent the standing wave pattern of the string, with the amplitude (0.023) and frequency (0.714 pi) determined by the initial conditions of the system.

a) To determine the harmonic, N, we can use the formula y(N) = 2y sin((Npix)/L)cos((vNtpi)/L), where L is the length of the string (7m) and v is the speed of waves in the string. Plugging in the given values, we get:

0.023 = 2(0.023) sin((Npi*7)/7) cos((vNtpi)/7)

Simplifying, we get:

1 = sin(Npi) cos((vNtpi)/7)

For a standing wave pattern, sin(Npi) must equal 1, so we can solve for N:

N = 1/pi

Therefore, the vibration corresponds to the first harmonic.

b) The frequency of the sound emitted by the string is equal to the frequency of the vibration, which is 0.714 pi milliseconds. To convert this to a more familiar unit of hertz (Hz), we can divide by 1000 to get:

f = (0.714 pi milliseconds)/1000 = 0.000714 pi Hz

c) The speed of waves in the string can be determined using the formula v = fλ, where f is the frequency and λ is the wavelength. The wavelength can be found using the equation λ = 2L/N, where N is the harmonic number. Plugging in the values, we get:

λ = 2(7m)/(1/pi) = 14pi m

Therefore, the speed of waves in the string is:

v = (0.000714 pi Hz)(14pi m) = 10 m/s

In summary, the vibration corresponds to the first harmonic, the frequency of the sound emitted is 0.000714 pi Hz, and the speed of waves in the string is 10 m/s.