Question

Identify which of the following wave functions describe(s) travelling wave(s).
(\(A_0, B_0, a\), and \(b\) are positive constants of appropriate dimensions)

• A. \(\psi(x,t) = A_0(x+t)^2\)
• B. \(\psi(x,t) = A_0 \sin(ax^2 + bt^2)\)
• C. \(\psi(x,t) = \frac{A_0}{B_0(x-t)^2 + 1}\)
• D. \(\psi(x,t) = A_0 e^{(ax+bt)^2}\)

Hint:

The defining characteristic of a 1D travelling wave is that the spatial and temporal dependence \((x, t)\) only appears in the combination \((x \pm vt)\) or \((kx \pm \omega t)\). If you can't write the function's argument as a linear combination of \(x\) and \(t\), it's not a travelling wave. Be aware that while some mathematical forms of travelling waves (like in A and D) are unbounded and may not represent physical energy-carrying waves, they are still considered travelling waves in a mathematical context.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A travelling wave is a disturbance that propagates through space while maintaining its shape. Mathematically, a one-dimensional travelling wave is described by any function whose argument is a linear combination of position \(x\) and time \(t\), typically of the form \((x \pm vt)\) or, more generally, \((ax \pm bt)\). The function describes the shape of the wave, and the argument describes its propagation.
Step 2: Key Formula or Approach:
The general form of a one-dimensional travelling wave is \(\psi(x,t) = f(ax \pm bt)\), where \(a\) and \(b\) are constants. We need to inspect the argument of each given function to see if it can be expressed in this form.
Step 3: Detailed Explanation:
(A) \(\psi(x,t) = A_0(x+t)^2\)
This function is of the form \(f(u) = A_0 u^2\), where the argument is \(u = x+t\). This is a linear combination of x and t. Specifically, it's of the form \(f(x+vt)\) with speed \(v=1\). This represents a parabolic pulse travelling in the negative x-direction. Therefore, it is a travelling wave. So, (A) is correct.
(B) \(\psi(x,t) = A_0 \sin(ax^2 + bt^2)\)
The argument of the sine function is \(ax^2 + bt^2\). The variables x and t are not in a linear combination. It is impossible to factor this expression into the form \(f(ax \pm bt)\). The points of constant phase, \(ax^2 + bt^2 = \text{constant}\), do not move with a constant velocity. This represents a form of standing wave or oscillation, but not a travelling wave. So, (B) is incorrect.
(C) \(\psi(x,t) = \frac{A_0{B_0(x-t)^2 + 1}\)}
This function can be written as \(f(u) = \frac{A_0}{B_0 u^2 + 1}\), where the argument is \(u = x-t\). This is of the form \(f(x-vt)\) with speed \(v=1\). It describes a pulse with a Lorentzian shape travelling in the positive x-direction. Therefore, it is a travelling wave. So, (C) is correct.
(D) \(\psi(x,t) = A_0 e^{(ax+bt)^2\)}
This function is of the form \(f(u) = A_0 e^{u^2}\), where the argument is \(u = ax+bt\). This is a linear combination of x and t. It can be written as \(f(a(x+(b/a)t))\), representing a pulse travelling in the negative x-direction with speed \(v = b/a\). Therefore, it is a travelling wave. So, (D) is correct.
Step 4: Final Answer:
The functions that describe travelling waves are those in options (A), (C), and (D), as they can all be expressed in the general form \(f(ax \pm bt)\).