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Wave Motion - Coggle Diagram
Wave Motion
Characteristics
Amplitude,
A
Wavelength, λ
Frequency,
f
Wave velocity,
v
Standing wave
E2 = (Emax/2) sin (kx + wt)
Etotal = Emax cos wt sin kx
E1 = (Emax/2) sin (kx - wt)
Nodes and Antinodes
Nodes
The positions of zero amplitude.
x=(n λ)/2
Half a wavelength (λ/2) separates two consecutive nodes
n=1, 2, 3…
Antinodes
The positions of maximum amplitude.
2asinkx = maximum [Maximum when sinkx=1]
x= (n+(1/2))( λ/2)
n=0,1,2,3,4 …
Type of waves
Transverse Wave
The motion of particles in a wave is perpendicular to the wave direction
Speed of travelling wave
velocity, v= ω/k
ω=angular frequency k=wave number
Example : Light
Longitudinal wave
The motion of particles in a wave is parallel to the wave direction
Example : Sound Wave
Principle of Superposition
Principle of superposition of waves describes how the individual waveforms can be algebraically added to determine the net waveform.
Condition
In Phase
y1(x, t) =a sin (kx – ωt).
y=y1+y2=2a sin (kx – ωt)
y2(x, t) =a sin (kx – ωt).
Completely Out of phase
y1(x, t) =a sin (kx – ωt).
y=y1+y2=0
y2(x,t) =a sin (kx – ωt+ π). y2(x,t)=-a sin(kx- ωt)
Partially Out of Phase, [φ>0 ; φ<π]
y1(x, t) =a sin (kx – ωt).
y=y1+y2=2a cos (φ/2) sin (kx – ωt + (φ/2))
y2(x, t) =a sin (kx – ωt+ φ).
Fourier's Theorem
Any complex periodic wave can be written as the sum of sinusoidal waves of different amplitudes, frequencies and phases.
Interference
Constructive Interference
Pure constructive interference produces a wave that has twice the amplitude of the individual waves, but has the same wavelength.
Destructive Interference
The disturbances are in the opposite direction for this superposition
The resulting amplitude is zero for pure destructive interference.
The waves completely cancel
Reflection and Transmission
Reflection of waves
Open boundary
When a wave strikes an interface in case of open boundary it will get reflected as well as refracted.
Take place without any phase change
Amplitude at the boundary is maximum
Superposition principle y= yi + yr =2a sin (kx – ωt)
Closed boundary / Rigid boundary
When a wave is incident on an interface it will completely get reflected.
Wave are completely out of phase at the boundary
The amplitude at the boundary is 0
yi(x, t) = a sin (kx – ωt)
yr(x, t) = a sin (kx + ωt + π) = – a sin (kx + ωt)
y= yi + yr =0
One dimensional wave equation
y(x,t) =Asin(kx-ωt+Φ)
Φ = Phase Constant
ω = Angular Frequency
k
= Wave Number
y(m)
= Maximum Amplitude
Formula : 1. k=2π/λ 2. ω=2π/T