Why are the products of sound wave lengths and frequencies always constant?

[dan1]

Hi everyone!
I was reading a textbook a while ago about waves and it had just finished talking about wave speeds and how it is the product of the frequency times the wave length. On the next page, it gave a table of frequencies and wave lengths of sound waves and how their products are all equal to 340 m/s. I suppose that makes sense: high and low notes at a concert produced at the same time would hit you at the same time. But here's my question: Why is this so? Why is sound always a constant velocity (assuming it is traveling through a consistent medium)?

 

[krd]

Because the speed of sound is constant in medium.

It's a wave. The frequency is how many times the wave cycles in a second. And the wavelength is the length of a cycle. When you multiply them you get a speed. And the speed will always be the speed of sound.

Frequency = cycles per second. Wavelength = length of a cycle. Multiply them and the answer is in metres per second. It's the speed of the wave.

 

[mikelepore]

For sound traveling through a solid or liquid, particles have electrical bonds with neighboring particles. These bonds are like bouncing springs, with well-defined time delays to return to ther equilibrium position when they have been temporarily deformed. Vibrations move from one particle to the next according to the spring characteristics of the bonds.

For sound traveling through a gas, the particles are independent projectiles that randomly collide with each other, but there is an average probability of such collisions, which depends on the temperature. This governs the transmission of a disturbance along some line.

 

[mikeph]

A mathematical answer would be that the dispersion relation of the wave equation for sound waves is linear, so the phase velocity is independent of the wave frequency (w/k = c for all w). It's a result of the wave equation.

Not all waves have to be like this- waves on a beach have a more complicated dispersion relation, and they will propagate with different speeds depending on their wavelength.

 

[harrylin]

Hi everyone!
I was reading a textbook a while ago about waves and it had just finished talking about wave speeds and how it is the product of the frequency times the wave length. On the next page, it gave a table of frequencies and wave lengths of sound waves and how their products are all equal to 340 m/s. I suppose that makes sense: high and low notes at a concert produced at the same time would hit you at the same time. But here's my question: Why is this so? Why is sound always a constant velocity (assuming it is traveling through a consistent medium)?

As others have indicated, but perhaps useful to emphasize: although the equation could be misinterpreted as such, the wave speed is not the result of the frequency and the wavelength. The source emits a wave with a certain frequency and the medium propagates it at a certain speed. The resulting wavelength is the quotient of the wave speed and the frequency.

See also as introduction:
https://en.wikipedia.org/wiki/Speed_of_sound#Basic_concept