ABSTRACT
The driving force acting on a length element dx, Eq.(15.24), is .10 →=α ( )2 2/ ,F x dx∂ Ψ ∂ so that the work done on this element, in moving it from α Ψ to ( ) ,dα α+ Ψ is ( )2 2/F x dxdα α− Ψ ∂ Ψ ∂ and the total work can be written as
1 2
dW F dx d F dx x x
α α⎛ ⎞ ⎛ ⎞∂ Ψ ∂ Ψ= − Ψ = − Ψ⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∫ 2 (17.1) Taking the sum of the contributions from all the elements in a finite section, one obtains
1 2
W v x
µ ⎛ ⎞∂ Ψ= − Ψ ⎜ ⎟∂⎝ ⎠∫ dx (17.2) where use has been of Eq.(15.26). Integrating by parts Eq.(17.2) and adding the kinetic energy in a finite section of the string gives
1 2
T d x
µ ∂Ψ⎛ ⎞= ⎜ ⎟∂⎝ ⎠∫ x (17.3) and this leads to
1 1 1 2 2 2
T W dx v dx v x x x
µ µ µ∂Ψ ∂Ψ ∂Ψ⎛ ⎞ ⎛ ⎞ ⎡ ⎤+ = + Ψ − Ψ⎜ ⎟ ⎜ ⎟ ⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠ ⎣ ⎦∫ ∫ ∂ (17.4) The last term on the right-hand side expresses the boundary conditions on the system, given by the end constraints. The second term on the right-hand side defines the potential energy stored in the same section
1 2
U v x
µ ∂Ψ⎛ ⎞= Ψ⎜ ⎟∂⎝ ⎠∫ dx (17.5) so that the total wave energy is given by
1, 2
E x t v dx t x
µ ⎡ ⎤∂Ψ ∂Ψ⎛ ⎞ ⎛ ⎞= +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦∫ (17.6) and the total energy density can be written as ( tx,ε )
( ) 2 221, 2
x t v t x
ε µ ⎡ ⎤∂Ψ ∂Ψ⎛ ⎞ ⎛ ⎞= +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ (17.7)
For travelling waves, which obey Eqs.(15.3) or (15.4), the two terms on the right-hand side which give the kinetic and potential energy densities are always equal. In calculating the energy of travelling harmonic waves we must not use the complex representation (16.16), since this would involve the product of two complex quantitites, whose real part is not equal to the product of the real parts of the two factors. Applying Eq.(17.7) to calculate the energy density carried by the harmonic wave (16.5), in the positive x-direction, we find that
( ) ( ) (2 2 2 2 21, sin2x t v k A kx t )ε µ ω ω ϕ= + − + (17.8) The instantaneous energy distribution, which travels along the x-axis at the same velocity as the wave profile, is illustrated in Figure 17.1. The kinetic and potential energy densities are equal at any point, the maxima of ( )tx,ε corresponding to those points where the disturbance is zero, whilst the values of both the velocity t∂Ψ∂ / and the slope
are the highest. x∂Ψ∂ / For a stationary wave (16.52) where the variables x and t appear in separate factors the kinetic and potential energy densities are no longer equal. Applying Eq.(17.7) gives
( ) ( ) ( ) ( ) ( )2 2 2 2 2 2 2 21, sin sin cos cos 2 n n n n n n n n n
x t A k x t v k k x tε µ ω ω ϕ ω⎡ ⎤= + +⎣ ⎦ϕ+ (17.9)
x
xΨ ( )x,t
ε (x,t)
Figure 17.1. Distribution of energy along a travelling harmonic wave As illustrated in Figure 17.2 for a stretched string fixed at both ends, when the string is straight, the energy (17.9) is all kinetic and located around the antinodes, whereas when the string is stationary, the energy is all potential and located around the nodes. Since energy passes between the nodes and the antinodes, there is no net flow of energy in one direction.
