A vibrating string

3.2.1 A vibrating string

Here we will examine the simplest second-order hyperbolic pde that arises as the governing equation for transverse waves on a vibrating string. The presentation is adapted from that found in Morse and Feshbach (1953). The solutions to this system are an instructive beginning for the more complex problems in two and three spatial dimensions. The one dimensional solutions can be considered as a basis for the common seismic processing step known as stretching that transforms a single seismic trace from time to depth or the reverse. Consider the situation shown in figure 3.1 where a snapshot (i.e. at constant time) of a vibrating string is shown. The displacement of the string from equilibrium, ψ(x) is shown as a dashed curve and the tension, T , always acts tangentially to this curve. The tension on both ends of a differential element of the string from x to x + dx creates a net force, F , that acts to restore the string to its equilibrium position. It is assumed that the magnitude of ψ(x) is sufficiently small at all points such that there is no significant change in T or in the length of the string. At position x in Figure 3.1, T (x) is directed tangentially along the string and has a component, T (x) sin θ, that acts to restore the string to it’s equilibrium position. (Here, theta is the angle between the string and the horizontal.) The assumption that T (x) is small implies that θ is

68 CHAPTER 3. WAVE PROPAGATION also small. This allows

∂ψ(x)

T sin θ(x) ≃ T tan θ(x) = T

∂x

where the last step follows from the geometric definition of the derivative. Considering the tension acting at both ends of the string element between x and x + dx allows the total restoring force on the element to be written as

∂ψ(x)

F (x)dx = T

where F (x) is the force per unit length and the minus sign in the square brackets is needed because the tension acts at x and x + dx in opposite directions.

Now, recall that the definition of the second derivative of ψ(x) is

dx→0 dx

∂x

∂x

Thus in the limit as dx becomes infinitesimal, equation (3.2) leads to

∂ 2 ψ(x)

(3.4) This important result says that the restoring force (per unit length) depends on the local curvature

F (x) = T

∂x 2

of the string. When the second derivative is positive, the curvature is concave (⌣) and when it is negative the curvature is convex (⌢). Since a positive force is directed upward in Figure 3.1 and a negative force is downward, it is easy to see that this force does act to restore the string to equilibrium.

If the string’s weight is negligible, then Newton’s second law (force = mass × acceleration) gives

∂ 2 ψ(x, t)

∂ 2 ψ(x, t)

2 ∂x =µ ∂t 2 (3.5) where µ is the mass per unit length and the dependence of ψ on both space and time is explicitly

acknowledged. It is customary to rearrange equation (3.5) to give the standard form for the one- dimensional scalar wave equation as

∂ 2 ψ(x, t)

1 ∂ 2 ψ(x, t)

where v =

µ is the wave velocity 3 This analysis has shown that the scalar wave equation arises for the vibrating string as a direct consequence of Newton’s second law. Wave equations invariably arise in this context, that is when a displacement is initiated in a continuum. Also typical is the assumption of small displacements. Generally, large displacements lead to nonlinear equations. The waves modelled here are known as transverse waves because the particle displacement is in a directional orthogonal to the direction of wave propagation. In section 3.2.2 longitudinal waves, that have particle oscillation in the direction of wave propagation, are discussed.

If there are external forces being applied to the string as specified by the force density 4 function

3 It is quite common to use velocity for this scalar quantity even though velocity is classically a vector quantity in physics. This book conforms with this common usage.

69 S(x, t), then equation (3.6) is usually modified to

3.2. THE WAVE EQUATION DERIVED FROM PHYSICS

∂ 2 ψ(x, t)

1 ∂ 2 ψ(x, t)

S(x, t)

Both equations (3.6) and (3.7) are examples of one-dimensional hyperbolic pde’s with constant coefficients. Equation (3.6) is said to be homogeneous while the presence of the source term, S(x, t), in (3.7) gives it the label inhomogeneous. Hyperbolic pde’s can usually be recognized right away because they involve the difference of spatial and temporal partial derivatives being equated to a source function. Wave equations can be second order in their derivatives as these are or any other order as long as the highest order of the time and space derivatives are the same. For example, a first order wave equation known as the advection equation is

∂φ(x, t)

∂φ(x, t)

where a is a constant. Exercise 3.2.1. Show that the left side of equation (3.6) can be factored into two operators of the

form of the left side of equation (3.8). What values does the constant a take in each expression. Show that ψ(x, t) = ψ 1 (x + vt) + ψ 2 (x − vt) is a solution to (3.6) byshowing that the two factors annihilate ψ 1 or ψ 2 . (ψ 1 (x+vt) means an arbitraryone-dimensional function of x+vt and similarly for ψ 2 (x − vt)). Can you explain the physical meaning of this result?