Lecture 6

• Sinusoidal waves: mathematical

• Waves on a string

• 2D/3D waves

• Sound and light

• Power and Intensity

Sinusoidal waves:mathematical

• Set wave in motion by x! (x" vt)

D(x, t) = A sin!2!

"x! !

tT

#+ "0

$

k = 2!" ; ! = "k

!0 sets initial condition:D(x = 0, t = 0) = A sin!0

wave number,

Example• A cork bobs on the surface of the water making an

oscillation every 3.0 s, with an amplitude of 1.0 m. Another cork 45.0 m away is observed to bob exactly 180 degrees out of phase with the first cork. What is the velocity of the water waves?

Waves on a string

• Newton’s laws applied to segment of string

(Fnet)y = may = (µ!x) ay

(Fnet)y = 2Ts sin ! ! "k2ATs!x

! v =!

Tsµ (independent of A/shape)

(evaluate slope of y = A cos(kx))

y(x, t) = A sin (kx! !t + "0)vy = !!A cos (kx! !t + "0)ay = !!2A sin (kx! !t + "0)

Example• A 2.31 kg rope is stretched between supports 10.4 m

apart. If one end of the rope is tweaked, how long will it take for the resulting disturbance to reach the other end? Assume that the tension in the rope is 42.8 N.

2D/3D waves

• 2D circular waves: wavefronts (lines locating crests), small section appear as straight lines far away

• 3D spherical waves...appear as planes far away, described by D(x, t) (same at every point in yz plane)

Phase and phase difference

• wavefronts are surfaces of same displacement constant phase

2D/3D waves D(r, t) = A(r) sin (kr ! !t + "0)

with A(r) decreasing with r

phase di!erence, "! = 2" !x!

"! = 2" between adjacent wavefronts(separated by #)

phase, ! = kx! "t + !0

D(x, t) = A sin !

Sound waves

• human ears: 20 Hz to 20 k Hz

• ultrasound: > 20 k Hz

vsound in air at 20! = 343 m/s(larger in liquid/solid)

Electromagnetic (EM) waves• oscillations of EM field, can travel in vacuum

e.g. light from stars

• visible spectrum: 400 nm (violet/blue to 700 nm (orange/red)

• EM spectrum: visible + higher frequencies (UV/X rays) + lower frequencies (IR/micro/radio waves)

• index of refraction (light slowed down):

frequency does not change (e.g., sound wave hitting water):

vlight = c = 3! 108 m/s in vacuum(" vsound)

! !sound " flight # fsound

n = speed of light in vacuumspeed of light in material = c

v

fvac.

!= c

!vac.

"= fmat.

!= vmat.

!mat.

"

! !mat. < !vac.

Power and Intensity• Power is rate of transfer of energy by wave

• Brightness/loudness depends also on area receiving power:

• Uniform spherical wave

intensity, I = Pa = power-to-area ratio

(SI units: W/m2)

I = Psource4!r2

(from energy conservation:total energy crossing wavefront is same)I1I2

= r22

r21

I ! A2

(energy of oscillations E = 12kA2)

Decibels (for wide range of human hearing)

• at threshold

• increasing by 10 dB I increases by a factor of 10

! = 10 dB log!

II0

"(dimensionless)

10!12 W / m2

! = 0

(threshold)

Example• The ``planet’’ Pluto’s average distance from the sun

is . . Calculate the sun’s intensity at the distance of Pluto, assuming the sun radiates a total power of W.

5.9! 1012 m

4.0! 1026