September 7 2026

I’m back with another update! Today, I learned about sound waves from HRK (physics olympiad textbook), which are pretty interesting but difficult to wrap my head around… of course, this is a study log, so I’m gonna put a little summary here.

Sound waves

Sound waves can be represented in a few different ways. Of these few, the ways presented in the book is pressure, density and displacement. We use the bulk modulus in relation to the pressure.

The Bulk Modulus

The bulk modulus is defined as:

$$B = -V \frac{\Delta V}{\Delta p}$$

1. Pressure and Density

For pressure as a sinusoidal wave: $p (x, t) = \Delta p_{m} \sin(kx - \omega t).$Similarly, for density as a sinusoidal wave:

$$\rho(x,t) = \Delta \rho_{m} \sin(kx-\omega t).$$

2. Displacement

For displacement as a sinusoidal wave:

$$s = \frac{\Delta p_{m}}{kB}\sin(kx-\omega t) = \frac{\Delta \rho_{m}}{k\rho_{0}} \sin(kx-\omega t).$$

Or more generally,

$$s = \frac{p(x,t)}{kB} = \frac{\rho(x,t)}{k\rho_{0}}.$$

I also learned about the speed of a longitudinal wave in a medium.

3. Speed of sound

The speed of a longitudinal wave can be expressed as:

$$v = \sqrt{\frac{B}{\rho}}$$

Woo! Simple enough! But then comes the parts that I struggle to memorize.

4. Power

We know that power can be expressed as $P = Fv$. Using this definition, we can do the following derivation:

$$F = pA = \Delta p_m \sin(kx-\omega t) A, v = \frac{ds}{dt} = \frac{\Delta p_m \omega}{kB} \sin(kx - \omega t)$$$$\implies P = \frac{A \omega (\Delta p_m)^2}{kB} \sin^2(kx-\omega t) = \frac{Av (\Delta p_m)^2}{B} \sin^2(kx-\omega t).$$

From this we can obtain an expression for $P_{avg}$. We use the fact that the average of $\sin^2(x)$ for some random x is $\frac{1}{2}$. Then, we get:

$$P_{avg} = \frac{Av (\Delta p_m)^2}{2B}.$$

Intensity

It is useful to define a value called Intensity as (Average Pressure)/(Area). This logarithmically links with how our ears react to sound.

$$I = \frac{P_av}{A} = \frac{v (\Delta p_m)^2}{2B}.$$

Now, we can define a quantity called the sound level SL, measured in decibels (dB) (Hmm I think I’ve seen this unit many times before…), which is defined as:

$$SL = 10 \log{\frac{I}{I_0}}$$

where $I_0 = 10^{-12} W/m^2$.

Finally, I learned about interference, and how it is important that we use pressure to represent sound because of this.

5. Interference

The phase difference can be expressed as, for two waves with the same wavelength:

$$\Delta \phi = \frac{2\pi}{\lambda} \Delta L.$$

When the pressure is in phase, they constructively interfere. When they are out of phase by $\pi$, they destructively interfere.

That’s all I learned for now! I still need to finish interference and learn all about how standing longitudinal waves work, or maybe review standing waves in general. It’s pretty cool cause I get to understand how my own instruments work! Then I’ll probably work through knzhou’s W1 handout. I’m probably gonna try Anki to learn all of this stuff in the short time that I have before my first selection test.