Waves Revision Notes PDF

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PHY1001: Summary of Waves....

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Waves Revision Notes November 2021

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Definition Definition of Waves A wave allows energy to be transferred from one point to another without any particle of the medium travelling between the two particles. There are two types of waves: • Transverse waves: propagated by vibrations perpendicular to the direction of travel • Longitudinal waves: propagated by vibrations parallel (in the same direction) to the direction of travel

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Describing a Wave Mathematically (Wave Equation)

The y (height) component of a waves motion is a function of x and t; y(x, t). We call this the wave function that describes the displacement of a particle at any time from equilibrium. The displacement equation is: y(x, t) = A sin (kx − ωx) k is wavenumber. k =

2.1

2π λ

=

2π v0 f

=

2πf v0

=

ω0 v0

so v0 =

ω0 k

Velocity and Acceleration

From the wave function (above) we can derive equations for velocity and acceleration.

Velocity and Acceleration from Wave Function y(x, t) = A sin (kx − ωt) Taking the partial derivative w.r.t t (i.e. we allow t to vary and x to stay constant since both are variables): ∂y(x, t) = −ωA cos (kx − ωt) vy (x, t) = ∂t The acceleration is found by taking the 2nd partial derivative w.r.t t: ay (x, t) =

∂2y = −ωA sin (kx − ωt) = −ω 2 y(x, y ) ∂t2

Notice the result above is the same as SHM (acceleration is −ω 2 times displacement) The wave equation is d2 y 1 d2 y = 2 2 2 v dx 0 dt

1

Waves

3

PHY1001

Principle of Superposition State the Principle of Superposition The resultant displacement at any point is equal to the sum of the separate displacements due to each wave.

y(x, t) = y1 (x, t) + y2 (x, t) The wave equation also holds for the superposition of two waves y1 and y2 d2 d2 (A1 y1 + A2 y2 ) = v20 2 (A1 y1 + A2 y2 ) 2 dx dt Combination of two identical waves: y=

  φ kx − ωt + 2

If v0 is constant we have non-dispersive waves. Perfect constructive interference (two identical waves): A′ = 2A Perfect destructive (two identical waves): A′ = 0

4

Beats

Frequency of beat:

1 fbeat = (f1 − f2 ) 2

Wave length of beat 1 λbeat

5

=

1 2



1 − 1 λ1 λ2

Dispersive Waves

Dispersive waves depend on frequency. Phase Velocity: v0 =

ω0 k0

vg =

dω dk

Group Velocity:

For two interacting waves: ω1 + ω2 k1 + k2 ω1 − ω2 vg = k1 − k2

v0 =

Normally dispersive vg < v0 Anomalously dispersive vg > v0

2



Waves

6 6.1

PHY1001

Doppler Effect Moving Source & Moving Observer fL =

v + vL f v + vS s

This is the most important equation for the Doppler Effect. The following to are really just special cases for vL = 0 and vS = 0. NB: if a source is moving towards you, you will hear a higher frequency (vS is taken as negative). For a source moving away from you vS is taken as positive and you hear a lower frequency.

6.2

Moving Listener (Observer)  vL  fL = 1 + fS v

6.3

Moving Source fS 1 + vvS  vS  λL = λS 1 + v fL =

3...