Propagating Waves on a String
How fast a wave runs along a stretched string and how much energy it carries. The speed \(v = \sqrt{T/\mu}\) from Newton's second law, a pulse climbing a hanging rope, the kinetic and potential energy of a piece of string, and the power a wave carries.
Why this chapter
The chapter Introduction to Waves described a travelling wave without asking what drives it. This chapter asks what the string itself does. Newton's second law, applied to one tiny piece of the string, gives the speed of every wave on it. Only two numbers enter: the tension and the mass per unit length. The same piece of string also holds energy, partly as motion and partly as stretch. That energy travels with the wave, so a wave on a string carries power from its source to the far end. These ideas return in every chapter on strings and sound that follows.
You will learn
- Derive the wave equation of a stretched string from Newton's second law, and read off the wave speed \(v = \sqrt{T/\mu}\).
- Find the wave speed on a string from its mass, length and tension. Explain why it does not depend on the frequency or the amplitude.
- Find the speed, the acceleration and the travel time of a pulse on a heavy hanging rope.
- Derive the kinetic and the potential energy of a piece of string, and show that they are equal for a travelling wave.
- Find the energy per unit length of a harmonic wave, its average \(\tfrac12\mu\omega^2A^2\) and the energy in one wavelength.
- Derive the power carried past a point, \(P = -T(\partial y/\partial x)(\partial y/\partial t)\), and the average power \(\tfrac12\mu v\omega^2A^2\) of a harmonic wave.
Before you begin
From the chapter Introduction to Waves you should know the travelling wave \(y = f(x - vt)\) and the wave equation. You should know the relation \(v_y = -v\,\partial y/\partial x\) between the particle velocity and the slope. You should know the harmonic wave \(y = A\sin(kx - \omega t)\), with \(v = \omega/k\). You should know Newton's second law, the work done by a force and the power \(P = Fv\), and the derivatives of sin and cos.
Lessons
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%Glossary
The key words of this chapter, with the meaning this chapter gives them.
- angular wave number
- The number \(k = 2\pi/\lambda\) in a harmonic wave, in rad/m. It tells how fast the phase changes along the wave.
- average power
- The power of a wave averaged over one period. For a harmonic wave on a string it is \(\tfrac12\mu v\omega^2A^2\).
- binomial approximation
- For a small number \(z\), \((1 + z)^n \approx 1 + nz\). With \(n = \tfrac12\) it gives \(\sqrt{1 + z} \approx 1 + \tfrac12 z\).
- energy per unit length
- The energy of a wave in a small piece of string divided by the length of the piece, \(dE/dx\), in J/m. For a harmonic wave its average is \(\tfrac12\mu\omega^2A^2\).
- harmonic wave
- A wave whose shape is a sine curve, \(y = A\sin(kx - \omega t)\). Each particle of the medium moves in simple harmonic motion.
- linear mass density
- The mass of a string per unit length, \(\mu = m/L\), in kg/m. A thick or heavy string has a large \(\mu\).
- partial derivative
- The rate of change of a quantity that depends on two variables, with one variable held fixed. For a wave \(y(x, t)\), \(\partial y/\partial t\) holds \(x\) fixed and \(\partial y/\partial x\) holds \(t\) fixed.
- potential energy of the wave
- The extra elastic energy stored in a string because the wave tilts and stretches its pieces beyond their length with no wave, \(dU = \tfrac12 T(\partial y/\partial x)^2 dx\).
- power of a wave
- The rate at which a wave carries energy past a point of the medium, in watts. It equals the rate at which the source supplies energy.
- tangent
- The straight line that touches a curve at one point and has the curve's direction there. The tension in a string acts along its tangent.
- tension
- The pulling force with which a stretched string acts on anything attached to it, and on its own neighbouring pieces. It acts along the string, in newtons.
- transverse speed
- The speed \(\partial y/\partial t\) with which one piece of a string moves up and down as a wave passes. It changes from moment to moment and differs from the wave speed.
- wave equation
- The equation \(\partial^2 y/\partial x^2 = (1/v^2)\,\partial^2 y/\partial t^2\), obeyed by every wave that travels at speed \(v\) without changing shape. For a string it reads \(T\,\partial^2 y/\partial x^2 = \mu\,\partial^2 y/\partial t^2\).
- wavelength
- The distance between two neighbouring points of a wave that are in the same state, such as two crests, written \(\lambda\).