# Wave propagation

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Wave propagation

Progressive (traveling) wave (前進波)

• Definition:
a progressive or traveling wave is a disturbance 干擾 that moves from a source to surrounding places, and as a result of which energy 能量 is transferred from one point to another.

Wave equation 波動方程

y(x,t) = A sin(ωt – kx)
represents the displacement of a particle at time t and at position x

y(x,t) = A sin(ωt – kx)
• A progressive wave propagating in the
positive x-axis direction.

• y: displacement 位移 • x: particle position 位置 • t: time • A: amplitude 振幅 • ω: angular speed 角速度, ω= 2πf • k: wave number 波數, 2π/λ

• The wave equation
y(x,t) =A sin(ωt – kx)
represents the displacement of a particle at time t and at position x.

• Wave speed v = ω/k • ω is related to f • k is related to wavelength λ

• sin(-θ) = - sinθ [t=0] • sin(π/2 – θ) = cos θ [t=π/4] • sin (θ- π/2) = - sin (π/2 – θ)[x=λ/4]
= - cos θ

• At t = 0,
y O 0

y(x,0) = A sin( kx) y(x,0) = A sin(kx)
t=0

P Q



x

• At point O, x = 0,
y 0

y(0,t) = A sin(ωt) x=0 T t

y 0 O P Q

t=0



x t = T/4

y 0

O

P Q



x O, x = 0

y 0 y 0 T T

t Q, x = /4 t

Rotating vector 旋轉矢量 diagram
y

The variation of displacements at the points O, P and Q with time can be represented by the projections 投影 on the y axis of the vectors rotating anti-clockwise 逆時針 with angular velocity , starting at the time as shown.



O P

Q

• y-x graph y

x 0 y • y-t graph P

t 0

At the point P

Huygen’s principle 惠更斯原理
• 1. any point on a wavefront 波陣面上的每

• 2. the new wavefront is the envelope of
these secondary wavelets 子波的包絡.

A

B ray

wavefronts

By Huygen’s principle, explain rectilinear propagation of light waves 光波是行直線

A

B

ray

• Explanation of law of reflection

• Explanation of law of refraction

1n2

= v1/v2 = λ1/λ2

Section 12.3 of further physics.

Reflection and phase changes 反射和相位變化 • When a wave is reflected at a denser medium「較

Fixed end C R

C

R

Applications of reflected waves
• Sonar & radar 聲納及雷達 • Long distance propagation of radio wave
(low frequency EM wave) by reflection from the ionosphere電離層.

ionosphere ground

Speed of mechanical waves 機械波
• Speed of transverse wave in a stretched string: • T: tension 張力 T • μ: mass per unit length v



• Speed of longitudinal waves in a solid rod: • E: Young modulus 楊氐模量 E • ρ: density 密度 of the rod v 



• Speed of wave depends on the elasticity 彈性
and density 密度 of the medium.

Phase difference (相差) between two particles having the same period of oscillation. From y-t graph,

y
Q D t t P

ΔΦ

y
P Q

ω

ΔΦ is the phase difference
ΔΦ/ 2π=Δt / T

• From y-x graph, we can compare their
distance apart with ΔΦ.
M’
y yM M ΔΦ

Dx

N ω

N

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