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Import converted Year 1 Semester 1 lecture notes from 443937a7d37850a3c57dfa863f35e80d07f2ccbc

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<b>Lecture 6: Standing Waves 2 Electric Boogaloo</b><br>
Thu 16 Oct 2025 13:00
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== Recap ==

<div class="pn-import-equation"><nowiki>$$\frac{\partial y}{\partial x} = 2 A k \cos(kx) \sin(\omega t)$$</nowiki></div>

<div class="pn-import-equation"><nowiki>$$\frac{\partial^2 y}{\partial x^2} = 2 A(-k^2) \sin (kx) \sin (\omega t)$$</nowiki></div>

<div class="pn-import-equation"><nowiki>$$\frac{\partial y}{\partial t} = 2 A \omega \sin (kx) \cos (\omega t)$$</nowiki></div>

<div class="pn-import-equation"><nowiki>$$\frac{\partial ^2 y}{\partial t^2} = 2A(-\omega^2)\sin (kx) \sin (\omega t)$$</nowiki></div>

Therefore:

<div class="pn-import-equation"><nowiki>$$\frac{\frac{\partial^2 y}{\partial t^2}}{\frac{\partial^2 y}{\partial x^2}} = \frac{2A(-\omega^2)\sin (kx) \sin (\omega t)}{2A(-k^2)\sin (kx) \sin (\omega t)} = \frac{\omega^2}{k^2} = v^2$$</nowiki></div>

<div class="pn-import-equation"><nowiki>$$\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$$</nowiki></div> Therefore a standing wave still obeys the wave equation, as it must.

== Standing Wave Properties ==

=== Wavelength ===

Consider a horizontal string from <nowiki>$x = 0$</nowiki> to <nowiki>$x = L$</nowiki>, with both ends fixed. We generate a sinusoidal wave pulse, which must satisfy: <div class="pn-import-equation"><nowiki>$$y(x, t) = 2A \sin(kx) \sin(\omega t)$$</nowiki></div>

We know at <nowiki>$x = L$</nowiki> and <nowiki>$x = 0$</nowiki>, <nowiki>$y = 0$</nowiki> at all times as fixed at this point. Therefore:

<div class="pn-import-equation"><nowiki>$$kL = n \pi, (n \in \mathbb{N})$$</nowiki></div>

For <nowiki>$n = 1$</nowiki>, we have half a wavelength on the string: <div class="pn-import-equation"><nowiki>$$\lambda = \frac{2L}{1} = 2L$$</nowiki></div>

And this has a general form: <nowiki>$\lambda = \frac{2L}{n}$</nowiki>.

=== Frequency ===

<div class="pn-import-equation"><nowiki>$$f_n = \frac{v}{\lambda_n} = \frac{v}{\left(\frac{2L}{n}\right)} = \frac{nv}{2L}$$</nowiki></div>

Crucially: <div class="pn-import-equation"><nowiki>$$f_1 = \frac{v}{2L}$$</nowiki></div> Is the first harmonic (or fundamental). <nowiki>$f_2 = 2f_1$</nowiki> is the second harmonic, or first overtone, etc. All of these, <nowiki>$f_n$</nowiki> where <nowiki>$n \in \mathbb{N}$</nowiki> are called “normal modes”. For each normal mode, the corresponding frequency is called the resonant frequency (natural frequency of the system).

What happens if we try to create a standing wave where <nowiki>$\lambda \neq \frac{2L}{n}$</nowiki>? In short, we cannot. The system will reject any attempts to do so.

=== Energy ===

Energy is proportional to <nowiki>$\omega^2$</nowiki>. Energy can only take certain discrete values (corresponding to <nowiki>$f_1, f_2, \ldots, f_n$</nowiki>), we find that the system has quantised possible values for energy.

To generate a wave with a higher frequency we either have to use a shorter L, or a higher v. A higher v is achieved by using a lighter string or placing the system under higher tension.

== Sound Waves ==

=== Notation ===

Displacement of a sound wave is denoted: <div class="pn-import-equation"><nowiki>$$s(x, t) = S_m \cos(kx - \omega t)$$</nowiki></div>

And pressure is given by: <div class="pn-import-equation"><nowiki>$$\Delta P(x, t) = \Delta P_m \sin(kx - \omega t)$$</nowiki></div>

== Different Boundary Conditions ==

The equations for standing waves given is only true for the boundary conditions of both ends fixed. If we vary these (for example left end fixed, right end not, wave initially travelling left) we get a different solution. For example, the first harmonic: <div class="pn-import-equation"><nowiki>$$L = \frac{1}{4 \lambda_1}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$f_1 = \frac{v}{4L}$$</nowiki></div> Where the left end forms a node (as required by boundary conditions) and the right end forms an antinode, as it is free to move. For the third harmonic: <div class="pn-import-equation"><nowiki>$$L = \frac{3}{4 \lambda_3}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$f_3 = \frac{3v}{4L} = 3 f_1$$</nowiki></div> And fifth: <div class="pn-import-equation"><nowiki>$$L = \frac{5}{4 \lambda_5}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$f_5 = \frac{5v}{4L} = 5f_1$$</nowiki></div>

Notably, this system cannot support even harmonics.

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Ash: Import converted Year 1 Semester 1 lecture notes from 443937a7d37850a3c57dfa863f35e80d07f2ccbc