# How does pitch change when sound velocity change?

The hot chocolate effect makes the pitch of sound of hitting a coffee mug lower when you stir the liquid within.

The explanation is that you lower the pitch because velocity lowers inside the bubbles you create when you stir the liquid.

What is the principle here? Velocity lowers inside bubbles therefore the frequency lowers (lower pitch)? What is the constant here?

• The wikipedia explanation (with context about predicting the speed of sound in a fluid) is: "When water is filled with air bubbles, however, the fluid's density is very close to the density of water, but the compressibility will be the compressibility of air. This greatly reduces the speed of sound in the liquid." Can you clarify whether this explanation is satisfactory, and if not, why not?
– rob
Jan 26, 2021 at 16:46
• I wanted to know basically, in the context of a simple formula, if the velocity changes and the frequency changes (I suppose, because pitch changes, is that correct though?) then what stays constant? @rob
– user137288
Jan 26, 2021 at 16:53

A wave's speed $$c$$, frequency $$f$$, and length $$\lambda$$ are related by

$$c = \lambda f$$

A standing wave occurs in a system with some length scale $$L$$, and generally fixes the wavelength to $$\lambda = aL$$, where $$a$$ is some rational number. For instance in a wave on a string the allowed wavelengths are $$\lambda = 2L/n$$ for positive integer $$n$$; for fluid in a half-open pipe the allowed wavelengths are $$\lambda = 4L/m$$ for odd positive integer $$m$$. Most introductory texts get lost in computational details and don't sufficiently emphasize the fundamental result that, for a standing wave, $$\lambda/L$$ is some constant.

As the link explains, the "hot chocolate effect" happens when many small air bubbles are introduced into a liquid, because the air bubbles dramatically change the fluid's compressibility without much effect on its density. That changes the wave's speed, which depends on both density and compressibility. Your question is which part of the wave's properties remain constant: it's the wavelength, which is related to the height of the liquid column in the cup.

• Interesting. So the wave that produces the sound is a water standing wave or a sound wave travelling in water/bubbles? Also isn’t the height related to amplitude rather than wavelength? (I know my interpretations are probably incorrect here, just looking to understand it better). Thanks.
– user137288
Jan 26, 2021 at 17:40
• Last question: the Wikipedia article says that you can repeat the process a few times after adding the soluble coffee/chocolate and after a few times it stops working because it reaches equilibrium. What does that mean? The bubbles stop forming because the soluble stuff gets completely dissolved?
– user137288
Jan 26, 2021 at 17:45
• (1) A standing sound wave in the water. Wikipedia suggests the fundamental is a quarter-wavelength from the bottom of the mug to the surface, like a half-open pipe; that approximation probably gets better as the liquid gets deeper, which is why an organ pipe has a nicer sound than a coffee mug. (2) The amplitude of a sound wave has units of pressure, not length. Another common beginner misconception, too complex for a comment. (3) I have never done this particular experiment, but I agree that if you put too much chocolate powder in your mug it just accumulates on the bottom.
– rob
Jan 26, 2021 at 18:45
• Thank you so much. As for (3) I actually meant that, after adding soluble powder, looks like you can perform the experiment a few times, after which it stops working. I suppose because the soluble powder stops creating bubbles after a while? I don’t know for sure.
– user137288
Jan 26, 2021 at 19:13
• I'm not sure either. This seems like the kind of thing that would be obsessively overanalyzed in the American Journal of Physics (which is why it's one of my favorite physics journals, though I don't currently have access). Lo and behold, that's what's in the Wikipedia bibliography, in 1982 and again in 1990.
– rob
Jan 26, 2021 at 19:27