#Waves # Intro - The Speed of Sound Sound is an integral part of us being people - we're able to hear and distinguish between different tones, using our minds to link each sound with an aspect of life. Whole disciplines, from audio technicians to physiologists, look at sound for guidance to fulfil a task - perhaps a speaker is making odd sounds or someone's voice has gone hoarse. However, as physicists, it's our job to make everyone's lives harder by generalising things through maths. Let's start! Simply put, a sound wave is a [[Waves Foundations#Oscillations and Wave Identities|longtitudinal wave]] caused by a compression of particles in an object, such as steel or water. When drawing sound waves, we're going to be using *rays* and *wavefronts* - as seen below: ![[Pasted image 20231001135215.png]] *A Poorly-Drawn Longitudinal Wave* Simply stated, the speed of sound is given by the equation: $v = \sqrt{ \frac{\tau}{\mu} }$ **Where:** - $\tau$ is the *elastic property of the wave*, a constant that denotes the ability of the wave to store potential energy - $\mu$ is the *inertial property of the wave*, another constant that denotes the ability of the wave to store kinetic energy Notice: this is really similar to the [[Waves Foundations#Wavespeeds of Strings|wavespeed of a transverse wave]], only this time we're using pressure gradients within the wave as a means of representing it! Now, I have no idea what this means. Take a look at the proof if you're willing to! >[!Danger]- Deriving the Wavespeed - Bulk Modulus Ver. >*Don't know what the bulk modulus is? Check out:* [[All About Elasticity (Mechanics)#Fluid Compression - Compressive Stress]]. > >Take a closer look at the diagram - we can imagine a compression to be similar to a 'pulse' on a transverse wave. This means that any object entering a compression (where there is a higher pressure) is going to experience a change in velocity: >$t = \frac{\Delta x}{v}$ >In a stretched spring, the potential energy of each 'pulse' comes from the elastic potential energy stored whenever the string stretches to become a pulse. Applying this to sound, the surrounding medium now stores the potential energy of the wave. >$B = \frac{\Delta p}{\Delta V / V}$ >**The Actual Proof:** > >With the groundwork set, let's start working towards our proof! Let's get another version of Newton's Second Law going: >$F = pA - (p+\Delta p)A$ >This secondary $p + \Delta p$ is used to define the change in the pressure of the medium during a rarefraction against a compression. This gives us: >$F = -\Delta pA$ >This means that the net force on a compression is directed towards the last rarefraction of the wave. There's a certain thing with potential energies we need to take into account! > >Notice that the velocity of the wave is going to require a velocity term to isolate. To do that, we'll enlist an artifact of our [[Bernoulli's Equation (Mechanics)|fluid section]], namely the identity $m = pV$. A little bit of maneuvering gives us: >$\Delta m = \rho \Delta V = \rho A \Delta x = \rho A v \Delta t$ >Since acceleration is $\frac{\Delta v}{\Delta t}$, we can get an equation for the total net force: >$-\Delta pA = \rho A v \Delta v$ >Rearranging gives us an advanced expression: >$-\frac{pA}{\Delta v A} = \rho v $ >$-\frac{p}{\Delta v} = \rho v$ >$-\frac{p}{\Delta v} = \rho v^2$ >$-\frac{p}{\Delta v / v} = \rho v^2$ >Notice that the bulk modulus revolves around a difference in the pressures of the air surrounding the pulse. Therefore, we can get: >$\frac{\Delta V}{V} = \frac{A\Delta v\Delta t}{Avt} = \frac{\Delta v}{v}$ >$\frac{p}{\Delta V / V} = \rho v^2$ >$\frac{B}{\rho} = v^2$ >This is how we get our final expression - the one HRW has: >$v = \sqrt{ \frac{B}{\rho} }$ >What a medley! # Sound Waves - The RIGID Definition Like transverse waves, sound waves are oscillations - they're just oriented on a different plane, similar to the directions of a magnetic and electric field. Since it still 'oscillates' - which [[Simple Harmonic and Pendulum Motion (Waves)|simple harmonic motion]] shows us is a sinusoidal, we're going to use said skills to define sound waves! Throwing another equation around, let's use this familiar sinusoidal as our base 'wave equation' of sorts. $s = s_{m}\cos(kx-\omega t)$ $s$ in this context refers to the *longitudinal displacement* - so the change in the forward displacement of the wave. If you haven't noticed already, here's the # Acoustic Wave Interference # Intensity and Power #