# How would velocity of sound, the fundamental frequency and wavelength of sound vary when the temperature of an organ pipe is increased?

Here is my approach to this:

neglecting any thermal expansion of the pipe:

By the Laplace formula for the speed of sound,

$$V=\sqrt{\frac{\gamma P}{\rho}}$$ where P is the pressure, $$\gamma$$ is the adiabatic constant and $$\rho$$ is the density of the medium.

Assuming the gas to be an ideal gas, we can use the ideal gas equation.
Hence we have:

$$V=\sqrt{\frac{\gamma RT}{M}}$$ where R is the gas constant, T is the absolute temperature, and M is the molar mass of air

So, when we increase the temperature, clearly, the velocity would increase as well. Coming to the fundamental frequency, we know

$$f_0$$ (fundamental frequency) $$\alpha$$ V (velocity of sound)

Hence, the fundamental frequency would also increase.

But how would we get the variance of the wavelength with temperature? I thought of using the relation

$$V=f\lambda$$ where V is the velocity of sound, f is the frequency and $$\lambda$$ is the wavelength.

According to which, the wavelength should also increase but according to my book, that's not right. Why does this happen?

• so thermal expansion of the organ pipe itself is neglected? Mar 30, 2021 at 13:39
• @sleepy yes it is. Mar 30, 2021 at 14:04