I don't understand what happens in reality (outside of wave theories). If I clap my hands I invest energy in the nearby air molecules, which move and transfer their energy to nearby molecules which move…and so on.
Why then, when I clap louder instead of investing more energy in the same particles making them move faster I simply involve more of them? Or am I wrong?
If I'm not, then what if I clap louder in a vacuum with limited air in it?
[Physics] Why do sound waves travel at the same speed moleculewise? (Same medium)
acousticswaves
Related Solutions
According to the MTU webpage Speed of Sound in Air, some things to consider:
if the ideal gas model is a good model for a real gas, then you can expect, for any specific gas, that there will be no pressure dependence for the speed of sound. This is because as you change the pressure of the gas, you will also change its density by the same factor.
However, the atmosphere is a mixture of chemicals with some varying, especially water vapour (hence, humidity), hence not an ideal gas - from the Wikipedia Speed of Sound page:
Sound speed in air varies slightly with pressure only because air is not quite an ideal gas.
This variation in speed of sound according to the MTU webpage is extremely minimal at most (see vertical scale below) for the range of atmospheric pressures at low humidity and a slight bit more pronounced at extreme humidity (see graph below):
Compositional changes (mainly water vapour) and especially air temperature has the greatest effect.
To expand on Xcheckr's answer:
The full equation for a single-frequency traveling wave is $$f(x,t) = A \sin(2\pi ft - \frac{2\pi}{\lambda}x).$$ where $f$ is the frequency, $t$ is time, $\lambda$ is the wavelength, $A$ is the amplitude, and $x$ is position. This is often written as $$f(x,t) = A \sin(\omega t - kx)$$ with $\omega = 2\pi f$ and $k = \frac{2\pi}{\lambda}$. If you look at a single point in space (hold $x$ constant), you see that the signal oscillates up and down in time. If you freeze time, (hold $t$ constant), you see the signal oscillates up and down as you move along it in space. If you pick a point on the wave and follow it as time goes forward (hold $f$ constant and let $t$ increase), you have to move in the positive $x$ direction to keep up with the point on the wave.
This only describes a wave of a single frequency. In general, anything of the form $$f(x,t) = w(\omega t - kx),$$ where $w$ is any function, describes a traveling wave.
Sinusoids turn up very often because the vibrating sources of the disturbances that give rise to sound waves are often well-described by $$\frac{\partial^2 s}{\partial t^2} = -a^2 s.$$ In this case, $s$ is the distance from some equilibrium position and $a$ is some constant. This describes the motion of a mass on a spring, which is a good model for guitar strings, speaker cones, drum membranes, saxophone reeds, vocal cords, and on and on. The general solution to that equation is $$s(t) = A\cos(a t) + B\sin(a t).$$ In this equation, one can see that $a$ is the frequency $\omega$ in the traveling wave equations by setting $x$ to a constant value (since the source isn't moving (unless you want to consider Doppler effects)).
For objects more complicated than a mass on a spring, there are multiple $a$ values, so that object can vibrate at multiple frequencies at the same time (think harmonics on a guitar). Figuring out the contributions of each of these frequencies is the purpose of a Fourier transform.
Best Answer
This is a very good question. I'm going to give you a more conceptual answer rather than the quick answer because I find this explanation helps my own students understand this better.
First, consider yourself standing in a gymnasium with a thousand people in it. Not a lot of room is there? Naturally, you'd want some personal space, so you push at the people near you, they have nowhere to go so they bump into the people near them and so on. Since everyone is pretty much shoulder to shoulder, the concussion wave you just produced travels only as fast as each person can stumble into the next one. But that didn't help you, everyone was still standing and you're still cramped. So you give everyone near you a giant shove. Now the people fall over, but you should notice that the wave still travels outwards at about the same speed. The reason is that everyone was still shoulder to shoulder and even though they're now falling over (giving you room to breathe), it takes the same amount of time for them to stumble and fall into the person right next to them.
That's the basic concept of the answer, but let's now talk about the air molecules when you clap. The idea that imparting more force from a stronger clap is correct. But consider in slo-mo what is happening when you clap. As your hands come down, they push air molecules out of the way, effectively imparting a net direction that each molecule moves in. This, in turn, may or may not make those initial molecules speed up, but that's not exactly relevant. After a very short distance, those molecules will hit others at arbitrary angles and impart that faster speed in many directions. The important thing to note is that your hand has moved those molecules into an area that already contained other molecules. Now each one is effectively in the gymnasium situation, all trying to push down their neighbours. To get around this, some of the excess molecules migrate to less crowded areas further away. But now those areas are super crowded and the cycle continues with the overall pressure wave traveling away from you.
So you see, you are absolutely correct, you do impart more energy and more velocity on the individual molecules. However, sound waves are not actually the molecules moving. The speed of sound depends on how long it takes the molecules to start feeling overcrowded and decide to move to a different place.
If you want the technical physics, the speed of an individual air molecule affects how fast it vibrates between regions of compression and rarefaction, which increases or decreases the pitch of the sound, but the molecule doesn't actually travel far from its original location. When you clap harder, most of the extra energy is spent moving more molecules. Slower hands move less molecules because the instantaneous velocity of each particle is much higher than that of your hand, so particles travelling away from the hand aren't moved by the clap. As the hand goes faster it catches up with the slower particles or the ones moving at angles to it, thus affecting more particles.
When the extra energy is used to speed up the molecules, It increases the pitch of the sound you'll hear, but the speed at which nearby molecules realize the increased pressure and start to flow away from it depends greatly on the overall density, temperature, etc of the air.
I probably should note that due to the increase in pressure and temperature very near to your hands, the speed of sound does technically increase right there. But as soon as the sound wave begins to move away from that area, it almost immediately returns to normal.