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Energy Transported by Waves

A wave is a disturbance that propagates through space and time with a transfer of energy.

From the discussion of wave propagation, it was seen that energy, E, is trasferred by vibrational energy as the wave travels through a medium.

E= 1/2 kA2    where A is the amplitude    ð The amount of energy that is transported is proportional to the square of the amplitude of the wave (A2)

I is defined as power (energy/time)/area = Power/area  in watts/m2    ð       I  is  proportional to A2  Note that E is in the numerator...

 

If the medium is the same in all directions, then the wave is spherical as shown on the right.

 

Surface Area, A, of a sphere = 4pr2       which increases for increasing r        

 

I = P/A = P/4pr2       ð    I is proportional to 1/r2

 

Using the 2 radii as shown on the right

 

I1 = P/4pr12        and      I2 = P/4pr22

I1/I2 = P/4pr12  / P/4pr22       = P/4pr12  (4pr22 /P)  =  r22/r12

I1/I2  =  r22/r12     ð I2 = I1 ( r12/r22 )

So, when the distance doubles, the intensity is reduced to 1/4 the original value

 

Example: Intensity at a distance of 1 = 2, what is intensity at a distance of 2?

 

I2 = I1 ( r12/r22 ) = (1)(1x1)/2*2 = 1/4


 

From earlier discussion,  I  is  proportional to A2  where A is the amplitude

Using similar reasoning as above

 

A2/A1 = r1/r2      ð  A2 = A1(r1/r2)        When the wave is twice as far from the source, the amplitude is 1/2 as large