Question

We know that the frequency and wavelength of an oscillation are related by the velocity of the wave (recall from above that !=%&). In standing waves, the wave velocities of the different harmonics are the same.Think about why this may be and explain that in your notebook. (Hint: The wave velocity is dependent on the static properties of the oscillating object like mass per unit length and tension)

Answer 1

Hey there!

As you said, the velocity of the wave can be given with the formula

where , f is the frequency and is the wavelength of the wave.

Since we are dealing with harmonics here, let me consider this as a sound wave, hence v¹= 343 m/s

We know that the frequency f is the inverse of the wavelength , (f = v¹ / , where v¹ is a constant here)

In that case, let us take the first harmonics. Here the wavelength is /2 for 1 second. Hence the frequency (number of waves in a second)

Substituting in we get

Hence the velocity of the first order harmonics is equal to the velocity of sound.

Considering Second harmonics,

Here the wavelength is for 1 second. Hence the frequency (number of waves in a second)

Substituting in we get

Hence the velocity of the second order harmonics is equal to the velocity of sound.

Considering third order harmonics

Here the wavelength is 3/2 for 1 second. Hence the frequency (number of waves in a second)

Substituting in we get

Hence the velocity of the third order harmonics is equal to the velocity of sound again.

The result would be the same for fourth harmonics with wavelength 2 , fifth harmonics with wavelength 5/2 ,... etc. For all substitutions with it's corresponding frequency the velocity obtained would be a constant.

I hope the explanation helps... Feel free to comment and discuss further... Cheers :)