phy1020.DVI

where is the density of the medium. Substituting this forminto Eq. (9.9), we have

ED^12 Svt!^2 A^2 : (9.10)

This says that the energyEcarried by the wave is proportional to the square of the amplitudeA, and also to
the square of the frequency!. ThepowerP(in watts) is the energy per unit time, or

PD

E

t

D^12 Sv!^2 A^2 : (9.11)

Now dividing the power by the surface areaSgives an expression for the waveintensityI(watts per square
meter):

ID

P

S

D^12 v!^2 A^2 : (9.12)

9.8 Wave Intensity

Another issue that often arises is how wave intensityIvaries with the distancerfrom the source of the waves.
The answer is: it depends upon the shape of the waves. The power emitted by the source will be distributed
along a surface at distancer, and the shape of that surface will depend on the shape of the waves.
One common case isspherical waves, which are produced by a point source of spherical source. For
spherical waves, the powerPemitted by the source is spread over the surface of a sphere of radiusr.If
the power is radiatedisotropically(that is, equally in all directions), then the intensity in any direction at a
distancerfrom the source will beIDP=.4r^2 /,soI/1=r^2. Since the intensity is proportional to the
square of the amplitude, this implies the waveamplitudedrops off asA/1=r. In summary, for spherical
waves,

I/

1

r^2

I A/

1

r

: (9.13)

Another case iscylindrical waves, which are produced by a line or cylindrical source. In this case the
power is distributed over the surface of a cylinder of radiusr, and we have

I/

1

r

I A/

1

p

r

: (9.14)

When either of these types of waves is observed very far from the source, they approximateplane waves,
where the wave fronts are planes. For plane waves, the intensityIand amplitudeAare both constant and
independent ofr:

IDconst:I ADconst: (9.15)

9.9 Ocean Waves

The speed of ocean waves is a function of their wavelength and the ocean depth. Ocean wave speed is given
by the expression^3

vD

s

g

2

tanh



2

d



; (9.16)

(^3) N. Mayo, “Ocean Waves—Their Energy and Power,”Physics Teacher, 35 , 352 (September 1997).