physics 1b03summer-lecture 9 wave motion sinusoidal waves energy and power in sinusoidal waves

Physics 1B03summer-Lecture 9

Wave MotionWave Motion

•Sinusoidal waves•Energy and power in sinusoidal waves


Sine Waves

f(x) = y(x,0) = A sin(kx)

For sinusoidal waves, the shape is a sine function, eg.,

Then y (x,t) = f(x – vt) = A sin[k(x – vt)]

A

-A

y

x

(A and k are constants)


Sine wave:

A

-A

y

x

v

(“lambda”)is the wavelength (length of one complete wave); and so (kx) must increase by 2π radians (one complete cycle) when x increases by . So k = 2, or

k = 2π / λ

y (x,t) = A sin[kx – t]


Rewrite: y = A sin [kx – kvt]=A sin [kx – t]

ω = 2πf

y = A sin [ constant – t]

“angular frequency”radians/sec

frequency: cycles/sec (=hertz)

The displacement of a particle at location x is a sinusoidal function of time – i.e., simple harmonic motion:

The “angular frequency” of the particle motion is =kv; the initial phase is kx (different for different particles).

Review: SHM is described by functions of the form y(t) = A cos(t+) = A sin(/2 – –t), etc., with


Quiz A

-A

y

x

Shown is a picture of a wave, y=A sin(kxt), at time t=0 .

a

b

cd

e

i) Which particle moves according to y=A cos(t) ?

ii) Which particle moves according to y=A sin(t) ?

iii) Sketch a graph of y(t) for particle e.


y(x,t) = A sin (kx ± t –)

angular wavenumberk = 2π / λ (radians/metre)

angular frequencyω = 2πf (radians/second)

amplitude “phase”

The most general form of sine wave is y = Asin(kx ± ωt – )

The wave speed is v = 1 wavelength / 1 period, so

v = fλ = ω / k

phase constant


Thus, y = A sin (kx ± ωt)

or

Note: Wave speed, v = 1 wavelength / 1 period

y = Asin(kx ± ωt )

A : amplitude

k = 2π/λ : “angular wavenumber” (radians/metre)

ω = 2πf : “angular frequency” (radians/second)

: “phase constant” (radians)

in general

v = f λ = ω/k


Transverse waves on a string:

(proof from Newton’s second law)

Electromagnetic wave (light, radio, etc.):

v = c 2.998108 m/s (in vacuum)

v = c/n (in a material with refractive index n)

T

wave lengthmass/unit

tension Fv

(proof from Maxwell’s Equations for E-M fields)

The wave velocity is determined by the properties of the medium:

Wave Wave VelocityVelocity


Exercise

What are and k for a 99.7 MHz FM radio wave?


Particle Particle VelocitiesVelocities

Particle displacement, y (x,t)

Particle velocity, vy = dy/dt (x held constant)

(Note that vy is not the wave speed v – different directions! )

Acceleration, 2

2

dt

yd

dt

dva y

y


y

tkxAdt

dva

tkxAdtdy

v

tkxAy

yy

y

2

2 )sin(

)cos(

)sin(

“Standard” sine wave:

maximum displacement, ymax = A maximum velocity, vmax = A maximum acceleration, amax = 2 A

Same as before for SHM !


ExampExamplele

string: 1 gram/m; 2.5 N tension

Oscillator:50 Hz, amplitude 5 mm

y

x

Find: y (x, t)vy (x, t) and maximum speed

ay (x, t) and maximum acceleration

vwave

physics 1b03summer-lecture 9 wave motion sinusoidal waves energy and power in sinusoidal waves

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