THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

4.4 General Solution of the Wave Equation 73in Figure 4.2, the difference between the y-components of the tension at the twoends of element ds is the net transverse force given by Fy = (T sin θ ) x+x − (T sin θ ) x (4.1)Here (T sin θ) x+x is the value of T sin θ at x + x, and (T sin θ) x is the valueat x. Letting x → dx and applying the Taylor’s series expansionf (x + dx) = f (x) + ∂ f (x)∂xdxEquation (4.1) can be rewritten as∂(T sin θ)dF y = (T sin θ ) x + − (T sin θ )∂xx =∂(T sin θ)dx∂xAs the displacement y is assumed to be small, θ will be correspondingly small andthe relationship sin θ ≈ tan θ applies, with tan θ equal to y/x. The net transverseforce on the element ds then becomes[(∂ T ∂y )]∂xdF y =dx = T ∂2 y∂x∂ x dx 2The mass of the string element is δ dx. Applying Newton’s law F = ma, we get:T ∂2 y∂x dx = y2 δdx∂2 ∂t 2Setting√Tc =δ(4.2)the equation of string motion becomes∂ 2 y∂t 2= c2 ∂2 y∂x 2 (4.3)The constant c defined in Equation (4.2) represents the propagation velocity of thetransverse wave. Equation (4.3) is the wave equation representing the wave disturbancespropagated along the string. This equation was first derived by LeonhardEuler in 1748.4.4 General Solution of the Wave EquationThe second-order partial differential Equation (4.3) has the general solutiony = f (ct − x) + g(ct + x) (4.4)where the functions, f (ct – x) and g(ct + x), are arbitrary with arguments (ct ± x).The first term of the right-hand side of Equation (4.4) represents a wave moving