1. Making Waves
Wednesday October 16, 2002

2. One dimensional wave equation
Example
Disturbance on a string y yo
T = tension (N)
Small amplitude
pulse x
µ = kg/m
Particles move vertically only – i.e. transverse to direction of wave
This is a transverse wave – particles move perpendicular to
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3. One dimensional wave
One can show that the disturbance (y = vertical co-ordinate of particle
in the string ) obeys the equation,
and
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4. Wave Equation The equation,
is an example of a one-dimensional wave equation.
It describes propagation in a non-dispersive media
i.e. velocity of propagation is the same for all frequencies
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5. Wave Equation Solutions are of the form x x
These solutions can be verified by direct substitution.
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6. Solution to wave equation
Let
Then
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7. Solution to the wave equation
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8. Solution to the wave equation
Hence
and similarly for g (x + vt)
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9. Interpretation
Represents a disturbance traveling along the positive x-axis
With velocity v – to see this let:
be a disturbance at x2 and time t2
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10. Interpretation Now at t1 = t2 – Δt we can write,
f (x2 – vt2) = f (x2 – v(t1+ Δt))
= f ( x2- vΔt – vt1)
= f ( x1- vt1)
i.e. the disturbance at x2 , t2 is exactly the same as it was
at x1 = x2 - v Δt, a time earlier.
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11. Interpretation y yo x v x1
vΔt x2
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12. The Nature of f and g
The functions f, g are completely general thus far
They represent a huge variety of possibilities – some of which cannot be written down in analytical form
i.e. there is no analytical description for many types of irregular disturbances
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13. Harmonic functions
Almost any type of disturbance can, however, be described approximately by a suitable combination of sines and cosines (i.e. in a Fourier series expansion)
Thus if we let f or g be a harmonic function (sin or cos), then we can provide an excellent description of almost any arbitrary disturbance with a suitable combination of sin and cos
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14. Harmonic Functions
Let’s begin by discussing harmonic functions with a single frequency
Harmonic – a given particle excecutes single harmonic motion (SHM)
Now if y = magnitude of disturbance, and the wave propagates in the positive x -direction
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15. Harmonic Functions Here the angular frequency ω = 2πf ; f = frequency
Фo = phase angle which is determined by initial condition (t=0)
Note that we could also write in terms of sin, with a different value of Фo.
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16. Harmonic Waves y At a particular position xo: yo t T
T = period – time required for one complete cycle of particle motion
yo = amplitude of wave – maximum displacement
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17. Harmonic Waves y At a particular time to: yo x λ
λ = wavelength – distance traveled in one period = vT = v/f
Or, λf = v – true for all monochromatic waves
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18. Harmonic motion λf = v ω = vk ; v = ω/k Definitions
(a) Wavenumber k = 2π/λ
ω = vk ; v = ω/k
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19. Harmonic motion λf = v Definitions (a) Phase of wave
The phase of a harmonic wave is simply the argument of the
sinusoidal function
where Фo = initial phase angle
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20. Harmonic Waves Order of x and t is unimportant in description of waves
kx ± ωt can be written ωt ± kx
Both describe the same wave – just different phase
But sign is important
kx – ωt and ωt – kx both describe +ive x dir’n
kx + ωt and ωt + kx both describe -ive x dir’n
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