Sound WavesPhysics 101

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Properties of Sound Waves

They propagate in three dimensions as opposed to the one dimensional waves we learned about previously

They cause the molecules of the medium to oscillate, creating alternating regions of higher and lower pressure.

Properties of Sound Waves

It helps to think of sound waves in terms of what we already know:

Sound is a longitudinal wave

The molecules in the medium (air) oscillate parallel (or antiparallel) to the direction of motion of the wave

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The figure represents the air molecules being displaced from one area to another

Compression & Rarefaction

When a sound wave passes through air it causes regions of higher pressure (compression) and lower pressure (rarefaction)

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!The medium is compressed above the normal pressure causing an increase in pressure

The medium is stretched which causes a lowering of pressure since the molecules become stretched apart

Note: Pressure variations are scalar and particle displacements are either parallel or antiparallel

Taken from Physics for Scientists and Engineers

The Speed of Sound

The speed of a sound wave depends on the properties of the medium through which it is propagating.

Usually we would calculate speed using :

v = velocity

Ts = Tension in the string/stiffness in the medium

µ = Linear mass density of the string/how much the spring is oscillating

BUT WAIT

The Speed of Sound

How can we determine how “stiff” the air is?

Instead, we have to use the equation

Where B (bulk modulus) =

The ratio of the change in pressure 𝛥p divided by the fractional change in the volume (𝛥V/V)

The negative sign indicates that the sign of 𝛥V/V is opposite the sign of 𝛥p.

Displacement Amplitude

Just like when we studied SHM we can also describe sound waves using a sinusoidal equation

s(x,t) = sm cos(kx- ωt +ɸ)

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Displacement Equation

s(x,t) = sm cos(kx- ωt +ɸ)

We use s to denote position

sm is the maximum displacement from equilibrium (the amplitude for sound waves)

k= (2π/λ). It is representative of the wave number

Remember: v= λ𝒇 = ω/k

ω= 2π𝒇= 2π/T. It is the angular frequency

ɸ represents the phase constant

Pressure in a Sound Wave

Not only do we describe position but we can also relate pressure to a sound wave

The variation in pressure is due to the increases and decreases in pressure from the ambient pressure

Ambient pressure: Pressure that comprises the gas in which a wave travels through

Pressure EquationThe equation for pressure in a sound wave is:

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In the pressure equation, B (bulk modulus) appears because it helps us relate volume change and pressure change.

There is a more specific description of how the equation is described on pg. 427 of the textbook but put simply, the equation for pressure is the derivative (with respect to x) of the displacement equation multiplied by the bulk modulus.

IntensityThe energy of a sound wave is the intensity (I)

I = wave of power delivered per unit area

I = P/A

P is the rate at which the wave delivers energy and A is the area that the wave is hitting

IntensityThe power of a mechanical wave is described as Pavg= (1/2)µvω2A2

For a sound wave, µ (with units kg/m) is replaced by ρ — mass density (with units kg/m3). This will give us units W/m2. A is replaced by sm since sm describes the amplitude of a sound wave

The equation becomes: where ω is the angular frequency, v is the wave speed and ρ is the density of the medium