Wave Mechanics Examples: Traveling-Wave Solutions of the 1D Wave Equation

This companion article provides exercises for WM15, wave mechanics: Traveling-Wave Solutions of the 1D wave equation. All exercises are stated first so they can be attempted without seeing the answers. Complete worked solutions follow in Part II.

The central WM15 results are

(1)

for sufficiently smooth profile functions. The first-order direction tests are

(2)

and a sinusoidal solution must satisfy

(3)

These results are standard consequences of the one-dimensional linear wave equation [1, 2, 3, 4, 7].

How to use this problem set

For every proposed wave, separate three questions: what direction and speed are implied by the argument; whether the function satisfies the PDE throughout the domain; and whether it also satisfies the initial and boundary data of the particular physical problem.

Part I: Exercises

Exercise 1: Read direction and speed from the argument

For each function, state the propagation direction and speed magnitude:

(a)	u = F(x − 6t),	(b)	u = G(x + 2.5t),	(4)
(c)	u = H(3x − 12t),	(d)	u = Q(4x + 20t).	(5)

For parts (c) and (d), first factor the argument into a constant times x ∓ ct.

Exercise 2: Verify the right-moving family

Let u(x,t) = F(x−ct) with F twice differentiable. Define ξ = x−ct. Derive u_x, u_xx, u_t, and u_tt by the chain rule and prove that u_tt = c²u _xx.

Exercise 3: Verify the left-moving family

Repeat Exercise 2 for u(x,t) = G(x + ct) using η = x + ct. Derive the first-order relation between u_t and u_x that identifies leftward propagation.

Exercise 4: Infer speed from two snapshots

The center moves from x₁ = 3.0 m at t₁ = 0.20 s to x₂ = 7.0 m at t₂ = 0.70 s.

1. Find Δt and Δx.
2. Determine the propagation speed.
3. Write a generic traveling-wave form using an arbitrary profile F.
4. State the corresponding first-order relation between u_t and u_x.

Exercise 5: Construct solutions for a specified PDE

For

(6)

find the wave speed; write general right- and left-moving solutions; give one explicit smooth right-moving Gaussian; and give one explicit left-moving sinusoid with k = 3 rad/m.

Exercise 6: Directly verify a Gaussian pulse

Test

(7)

against u_tt = 16u_xx. Carry out the derivatives explicitly rather than appealing only to the general theorem.

Exercise 7: Derive the sinusoidal solution condition

For

(8)

compute u_xx and u_tt and derive the condition required by u_tt = c²u _xx. Then take c = 8 m/s and k = 2.5 rad/m and determine ω and the ordinary frequency f.

Exercise 8: Read the ω-k solution test

For u_tt = 25u_xx, the figure shows the allowed line and two candidate sinusoidal waves.

Candidate A has (k,ω) = (4, 20) and candidate B has (k,ω) = (4, 12) in SI-compatible angular units. Decide which candidate satisfies the PDE, find the speed implied by candidate B, and explain why agreement only at isolated zeros of the cosine would not be sufficient.

Exercise 9: Direction from first-order derivatives

At one event a single shape-preserving traveling wave has

(9)

Determine the propagation direction and speed. Repeat if u_t changes to +1.50 m/s while u_x is unchanged.

Exercise 10: Read characteristic lines

Identify which line corresponds to x − ct = constant and which to x + ct = constant. Explain why the second-order PDE supports both directions. If a right-moving line is x = 1.2t − 3, find its speed.

Exercise 11: Superpose right- and left-moving solutions

Let u₁ = F(x − ct) and u₂ = G(x + ct), with both profiles twice differentiable. Use linearity to prove that

(10)

also satisfies the wave equation. Explain why this does not yet determine F and G for a particular initial-value problem.

Exercise 12: Propagate a prescribed initial shape

A right-moving pulse obeys u_tt = 100u_xx and at t = 0 has

(11)

Find the speed, write u(x,t), write the profile at t = 0.30 s, and determine how far the center has moved.

Exercise 13: Smoothness and the classical solution requirement

Compare u₁ = e^−(x−ct)2 with u ₂ = |x − ct|. Which is a classical twice-differentiable solution everywhere? Along what line does the cusp fail the WM15 smoothness assumption? What are u_2,xx and u_2,tt away from that line?

Exercise 14: Synthesis - mechanics to traveling solution

An ideal string has

(12)

A right-moving sinusoid has wavelength λ = 1.50 m and amplitude A = 4.0 mm. At t = 0 it is at maximum positive displacement at x = 0.

Find c, k, ω, and f; write a right-moving sinusoidal solution; verify μu_tt = Tu_xx; and write the corresponding left-moving sinusoid.

Part II: Complete Worked Solutions

Solution 1: Read direction and speed from the argument

Use F(x − ct) for rightward motion and G(x + ct) for leftward motion. Factoring the arguments when necessary gives

F(x − 6t)	: +x, c = 6 m/s,	(13)
G(x + 2.5t)	: −x, c = 2.5 m/s,	(14)
H(3x − 12t) = H3(x − 4t)	: +x, c = 4 m/s,	(15)
Q(4x + 20t) = Q4(x + 5t)	: −x, c = 5 m/s.	(16)

The prefactor multiplying the whole argument changes its spatial scale but not the translation speed.

Solution 2: Verify the right-moving family

Let ξ = x − ct and u = F(ξ). Since ξ_x = 1 and ξ_t = −c, the chain rule gives

ux	= F′(ξ),	uxx	= F′′(ξ),	(17)
ut	= −cF′(ξ),	utt	= c2F′′(ξ).	(18)

Therefore

(19)

The first time derivative also gives the directional relation u_t = −cu_x.

Solution 3: Verify the left-moving family

Let η = x + ct and u = G(η). Here η_x = 1 and η_t = c, so

ux	= G′(η),	uxx	= G′′(η),	(20)
ut	= cG′(η),	utt	= c2G′′(η).	(21)

Thus

(22)

The plus sign in the first-order relation distinguishes the left-moving family in this convention.

Solution 4: Infer speed from two snapshots

The measured differences are

(23)

Hence

(24)

Because the motion is toward increasing x, a generic form is

(25)

Solution 5: Construct solutions for a specified PDE

From u_tt = 49u_xx,

(26)

Therefore the two traveling families are

(27)

One smooth Gaussian example is u_R = e^−(x−7t)2. For a left-moving sinusoid with k = 3 rad/m, the required angular frequency is ω = ck = 21 rad/s, so one example is

(28)

Solution 6: Directly verify a Gaussian pulse

Set ξ = x − 4t and write u = 2e^−ξ2. Differentiating the profile with respect to ξ gives

(29)

Because ξ_x = 1 and ξ_t = −4,

(30)

Hence

(31)

so the Gaussian is a classical right-moving solution.

Solution 7: Derive the sinusoidal solution condition

For u = A cos(kx − ωt + ϕ),

(32)

Substitution into the PDE requires ω² = c²k². For positive ω, k, and c,

(33)

With c = 8 m/s and k = 2.5 rad/m,

(34)

Solution 8: Read the ω-k solution test

The PDE u_tt = 25u_xx requires c = 5 m/s and therefore ω = 5k.

Candidate A gives ω∕k = 20∕4 = 5 m/s, so it satisfies the PDE. Candidate B gives ω∕k = 12∕4 = 3 m/s, so it does not.

(35)

At isolated zeros of the cosine, both sides of a mismatched PDE can vanish accidentally. A solution must satisfy the equation throughout the relevant domain, not just at selected points.

Solution 9: Direction from first-order derivatives

For a right-moving profile, u_t = −cu_x. The given values have opposite signs, so

(36)

Thus the wave moves toward +x. If u_t = +1.50 m/s while u_x remains positive, then u_t = +cu_x and the wave moves toward −x with the same speed magnitude.

Solution 10: Read characteristic lines

A feature with x − ct = constant moves toward increasing x as t increases; a feature with x + ct = constant moves toward decreasing x.

The second-order PDE contains c², so the directional sign disappears after two time derivatives. Both propagation directions therefore satisfy the same PDE.

For x = 1.2t − 3, the coefficient of t is the speed:

(37)

in the corresponding distance-per-time units.

Solution 11: Superpose right- and left-moving solutions

Each component satisfies the PDE:

(38)

For u = u₁ + u₂, linearity gives

utt	= (u1)tt + (u2)tt	(39)
	= c2(u 1)xx + (u2)xx	(40)
	= c2u xx.	(41)

Therefore

(42)

is also a solution. The PDE alone does not determine the specific functions F and G; initial and boundary data are needed to select them.

Solution 12: Propagate a prescribed initial shape

The PDE gives c = = 10 m/s. The initial profile is F(x) = 0.010∕(1 + x²). Since the pulse is known to move entirely toward +x,

(43)

At t = 0.30 s,

(44)

The center has moved ct = (10)(0.30) =

3.0 m .

Solution 13: Smoothness and the classical solution requirement

The Gaussian e^−(x−ct)2 is smooth and has ordinary second derivatives everywhere. The cusp |x−ct| fails to be differentiable in the ordinary sense along

(45)

Away from that line, the absolute-value profile is locally linear, so

(46)

At the cusp itself the classical second derivatives are not defined. Thus WM15’s statement about arbitrary traveling profiles must be read as “arbitrary sufficiently smooth traveling profiles.” More advanced PDE theory can treat nonsmooth profiles with generalized notions of solution.

Solution 14: Synthesis - mechanics to traveling solution

For an ideal string,

(47)

With λ = 1.50 m,

(48)

Hence

(49)

The phase condition u(0, 0) = +A is satisfied by zero phase constant, so the right-moving wave is

(50)

For a sinusoid, u_xx = −k²u and u _tt = −ω²u. Since ω² = (T∕μ)k²,

(51)

Therefore the function satisfies the mechanical string equation. The corresponding left-moving wave is

(52)

Common mistakes

• Reading H(3x − 12t) as a 12 m/s wave instead of first factoring the argument.
• Losing the minus sign in the first time derivative of F(x − ct).
• Carrying that minus sign into the second time derivative, where two factors of −c produce +c².
• testing a sinusoidal candidate at only one point instead of throughout the domain.
• Assuming the second-order wave equation by itself chooses a propagation direction.
• Treating a cusp exactly like a smooth classical solution.
• Forgetting that satisfying the PDE does not automatically satisfy a particular problem’s initial or boundary conditions.

What WM15E1 reinforces

The main verification result is

(53)

for sufficiently smooth profiles. Direction is retained by the first-order relations, while the second-order wave equation supports both directions. For sinusoidal waves the same requirement becomes ω = ck.

The next stage can therefore ask the stronger question: given initial data, how are the two traveling profiles F and G determined?

References