16.1 Propagation of a Disturbance

16.1 Pulse traveling to the right y (x, t) = ƒ(x – vt)

16.2 Pulse traveling to the left y (x, t) = ƒ(x + vt)

16.2 The Traveling Wave Model

16.3

16.4

16.5

16.6

16.7

16.8 Angular wave number

1ƒ

T=

y(x,t) ≡Asin

2πλ

x−vt( )⎡

⎣⎢

⎤

⎦⎥

y(x,t) ≡Asin 2π x

λ−tT

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥

y(x,0) =Asin

2πλx

⎛

⎝⎜⎞

⎠⎟

v =

∆x∆t

=λT

k ≡

2πλ

16.2 The Traveling Wave Model, cont.

16.9 Angular frequency

16.10 Wave function for a y = A sin (k x – t)

sinusoidal wave

16.11

16.12 Speed of a sinusoidal wave v = λƒ

16.13 General expression for a y = A sin (k x – t + ) sinusoidal wave

22 ƒ

T

π π= =

v =

k

16.2 The Traveling Wave Model, cont.

16.14

16.15

16.16 vy, max = A

16.17 ay, max = 2A

v

y=dydt⎤

⎦⎥x=constant

=∂y∂t

=−Acos kx−t( )

ay=dvydt

⎤

⎦⎥⎥x=constant

=∂vy∂t

=− 2Asin kx−t( )

16.3 The Speed of Waves on Strings

16.18 Speed of a wave on a

stretched string v =

Tμ

16.5 Rate of Energy Transfer by SinusoidalWaves on Strings

16.19 dK = ½ (μdx) vy2

16.20 Eλ = Uλ + Kλ = ½μ2A 2λ

16.21 Power of a wave ℘ =

12

μ 2A2v

16.6 The Linear Wave Equation

16.22

16.23

16.24

F

y∑ ≈T(tanθB −tanθA)

Fy∑ ≈T

∂y∂x

⎛

⎝⎜⎞

⎠⎟B−

∂y∂x

⎛

⎝⎜⎞

⎠⎟A

⎡

⎣⎢⎢

⎤

⎦⎥⎥

F

y∑ =may =µ∆x∂2y∂x2

⎛

⎝⎜⎞

⎠⎟

16.6 The Linear Wave Equation, cont.

16.25

16.26 Linear wave equation for a

string

16.27 Linear wave equation in

general

( ) ( )2

2B A

y x y xy

T t x

μ ∂ ∂ − ∂ ∂∂=

∂ Δ

2 2

2 2

y y

T t x

μ ∂ ∂=

∂ ∂

2 2

2 2 2

1y y

x v t

∂ ∂=

∂ ∂