September 8 2026

T - 4 days. I’m cooked :(

As usual, I continued with sound part 2 today, learning about standing sound waves + doppler effect. Here’s a quick summary for today:

Sound Waves (part 2)

Standing Sound Waves

Imagine you had a tube. An open end represents a node, as the outside pressure remains the same. This acts like a fixed end in a string for reflection. Note that in an open end the actual wavelength is slightly extended. On the other hand, a closed end represents an antinode, like an open end of a string for reflection. We don’t really have to make a correction for wavelength here.

Here are conditions for each:

Same-typed end tube

Where L is the length of the tube, standing waves act in the pattern of:

$\lambda = \frac{2L}{n}$

where $n \in \mathbb{Z}$.

Open end to closed end tube

Where L is the length of the tube, standing waves act in the pattern of:

$\lambda = \frac{4L}{n}$

where $n \in {1, 3, 5, \dots}$.

When systems are capable at vibrating at a number of frequencies, the lowest frequency is called the fundamental frequency and the higher frequencies are called overtones. When the overtones are integer mulitples of the fundamental, these overtones are called harmonics.

Beats

When two sound are being played with a similar frequency at the same time, a phenomenon called “Beats” occurs. For convenience, we use the same amplitude here, but it works with different amplitudes as well.

Define $\Delta p_1(t)$ as $\Delta p_m \sin(\omega_1 t)$ and $\Delta p_2(t)$ as $\Delta p_m \sin(\omega_2 t)$

We can add them together using the sum to product identity:

Sum-to-product identity

$\sin(a) + \sin(b) = 2\cos(\frac{a-b}{2})\sin(\frac{a+b}{2})$

With this, we can obtain:

$\Delta p_1 + \Delta p_2 = (\Delta p_m \cos(\frac{\omega_1 - \omega_2}{2} t))\sin(\frac{\omega_1 + \omega_2}{2} t).$

If there is little difference between $\omega_1$ and $\omega_2$, then the frequency of the beats becomes lower making them more obvious to hear.

The frequency of the beats are related to double the value of $\frac{\omega_1 - \omega_2}{2}$.

Using this, we obtain the value of frequency as $|f_1 - f_2|$. (Actually I should have used absolute sign earlier but whatever). Note that the absolute value makes it so two different frequencies can have the same frequency of beats.

The Doppler effect

For this we can cover two cases.

Case 1

Observer is moving towards source.

The source receives $\frac{v_o t}{\lambda}$ extra waves. This gives the number of waves you receive as:

$$n = \frac{v_0t}{\lambda} + \frac{vt}{\lambda}$$

The frequency is the number of waves per unit time. We can directly see that the new frequency is as follows:

$$f' = \frac{v_0 + v}{\lambda} = \frac{v_0 + v}{v/f} = f \frac{v_0 + v}{v}.$$

It’s easy to see that if the observer is moving away then we can replace the plus sign with the minus.

Case 2

Source is moving towards the observer.

To solve this, we can find the apparent wavelength: when the wave travels by one full wavelength, the time taken is $t = \lambda/v$. But the source moves another $v_s t$ meters. We find $\lambda' = \lambda - v_s t = \lambda - v_s \frac{\lambda }{v} = \lambda\frac{v - v_s}{v}$.

Using this result, it’s easy to solve for the frequency, which is speed/wavelength, giving:

$$f' = \frac{v^2}{\lambda(v - v_s)} = f \frac{v}{\lambda(v-v_s)}.$$

Again, like before, if the source is moving away then we can replace this minus sign with a plus.

We can finally combine these results to get this goofy ahh equation:

$$f' = f \frac{v\pm v_o}{v \mp v_s}.$$

This is quite an elegant equation!

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Currently it is 12:46 am the next day… I really need to get my sleep, so I’m gonna go now. I was really locked out by watching too much youtube - from tomorrow, I’m aiming to become much more efficient. Not studying for 8 hours, but not wasting the entire day watching youtube. Hopefully I can get some things done tomorrow!