What experimental evidence shows that sound velocity is the same for all wavelengths? I'm studying sound waves with Halliday's book, and after reading the whole chapter, one of the questions suggested was:

What is the experimental evidence that allows the assumption that the sound wave velocity in air is the same for all wavelengths?

I was surprised that such an apparently simple question really caught me. I can't seem to think of a better answer than

"The fact that we make experiments and see that the velocity is the same for all wavelengths".

But of course there must be a better answer, this one is just Why?? Because yes.
What would be a better answer to that question?
 A: It's a bit pat maybe, but if the wavelength-dependence were detectable over human distance scales, the (quality of) sound (not just the volume) of music, speech etc. would depend noticeably on how far one were from the source.
A: Good theoretical answer is that it results from linear acoustical wave equation and its presuppositions. It is therefore good approximation whenever linear acoustics still can describe the wave propagation (that would by e.g. 90% of room acoustics practical examples). Typical examples of problematic models are large-amplitude events (e.g. a shockwave after an explosion) or highly dispersive media (e.g. mechanical wave on a stiff string).
There is probably no chance to prove it theoreticaly on a high-school level. 
AGML's answer is very good. Let me propose an experiment. Go somewhere outside, place the microphone at least 50 m from you and clap your hands loudly (or fire a starter's gun etc.). Why? Because these almost pulse sounds have almost flat spectrum (i.e. really many wavelengths are present). Then listen to the recorded sound. If you hear clapping (or firing etc.), i.e. signal wasn't modified, you certainly must have the constant speed of sound for at least many frequencies.
A: The spectrum of various resonant tube arrangements (half-open, fully-open fully closed) is something that can be measured in a very basic laboratory and gives solid evidence that the claim is true over the kinds of frequencies that are accessible  in such a lab. Say a few hundred to a few thousand hertz.
