Wave Mechanics Examples: Translating Disturbances

This companion article provides exercises for WM06, wave mechanics: Translating Disturbances. The emphasis is on interpreting and constructing shape-preserving disturbances of the forms

(1)

and

(2)

The exercises remain deliberately within the WM06 scope. They use arbitrary translated shapes, feature tracking, propagation speed, spatial snapshots, and time histories. The sinusoidal traveling-wave form A cos(kx − ωt + ϕ) is reserved for WM07.

How to use this problem set

Attempt every exercise in Part I before consulting Part II. For sign questions, do not rely only on memory. Track a fixed feature by setting the function argument equal to a constant. For numerical problems, write units at each step and distinguish propagation of the disturbance from motion of the material medium.

Part I: Exercises

Exercise 1: Direction and speed from the translated argument

For each disturbance, state the direction of propagation and the speed.

1. u(x,t) = F(x − 4t), with x in meters and t in seconds.
2. u(x,t) = G(x + 7t).
3. u(x,t) = H(x − 0.25t).
4. Explain why the sign inside the argument is opposite the direction of motion.

Exercise 2: Track a fixed feature

A right-moving disturbance is

A recognizable feature corresponds to the fixed argument value

1. Write the equation that gives the position of this feature as a function of time.
2. Find its position at t = 0, 1 s, and 4 s.
3. Compute its displacement over the four-second interval.

Exercise 3: Left-moving feature

A disturbance is

A marked feature is located at x = 8 m when t = 0.

1. Find the feature position after 1.2 s.
2. Find the time at which the feature reaches x = −7 m.
3. State the propagation direction and speed.

Exercise 4: Read pulse translation from snapshots

The following graph shows the same localized disturbance at three times.

1. In which direction is the pulse traveling?
2. What distance does the pulse center travel during each 1 s interval?
3. Determine the propagation speed.
4. If the center is at x = 4 m when t = 2 s, where will it be at t = 5.5 s?

Exercise 5: Construct translated disturbances

At t = 0 a disturbance has shape F(x).

1. Write the function if the shape moves toward +x at 6 m/s.
2. Write the function if the same shape moves toward −x at 6 m/s.
3. At t = 2 s, by how far has each profile shifted from its initial location?
4. Does changing the sign alter the shape F itself?

Exercise 6: Recover a snapshot

A disturbance is given by

1. Write the snapshot u(x, 0).
2. Write the snapshot u(x, 3).
3. Describe geometrically how u(x, 3) is related to u(x, 0).
4. Write the snapshot at a general time t = t₀.

Exercise 7: Dimensional consistency

Assume x is measured in meters and t in seconds. Decide whether each expression could be a dimensionally valid translated disturbance. Explain.

1. F(x − 4t), where 4 carries units of m/s.
2. F(x − 4), where 4 carries units of m/s.
3. F(x − 3t²), where the coefficient 3 has units of m/s².
4. F(x − vt), where v has units of m/s.

Exercise 8: Gaussian pulse

Consider

with A = 5.0 mm, c = 2.0 m/s, and a = 0.50 m.

1. In which direction does the pulse move?
2. Where is the pulse maximum at time t?
3. Where is the maximum at t = 3.0 s?
4. Evaluate the displacement at a point one width a to the right of the center. Express the answer as a multiple of A and numerically in millimeters.

Exercise 9: Spatial snapshot versus time history

The following pair of graphs refers to the right-moving pulse

1. Which graph is a spatial snapshot and which is a time history?
2. At t = 1 s, where is the pulse center?
3. At what time does the pulse center reach the observer at x = 5 m?
4. Explain why the two graphs describe the same disturbance but use different horizontal variables.

Exercise 10: Propagation versus material motion

A transverse pulse travels to the right along a stretched string.

1. Does a marked material point on the string have to move to the right with the pulse?
2. What kind of local motion can the string element undergo while the disturbance travels horizontally?
3. Distinguish propagation speed from material-element velocity in words.
4. Why is this distinction important when later discussing transport of energy?

Exercise 11: Read an x-t diagram

The following plot tracks two recognizable features.

1. Which trajectory corresponds to motion toward +x?
2. Which corresponds to motion toward −x?
3. State the velocity represented by each line.
4. For the line x = 1 + 2t, find the position at t = 4 s.
5. For the line x = 4 − t, find the time when the feature reaches x = 0.

Exercise 12: Diagnose a sign error

A student writes:

“The disturbance F(x−ct) must move left because the minus sign points toward negative x.”

1. Show mathematically why the statement is incorrect by tracking a fixed argument ξ₀.
2. Give the correct direction of F(x − ct).
3. Give the correct direction of F(x + ct).
4. State a reliable procedure for determining direction without memorizing a sign rule.

Exercise 13: Shape preservation and its limits

A profile is observed to become wider as it propagates.

1. Can the entire evolution be represented exactly as F(x − ct) with one fixed function F? Explain.
2. What property of F(x − ct) prevents the profile from broadening?
3. Name two physical effects mentioned in WM06 that may cause a real disturbance to change shape.
4. Why is the shape-preserving model still useful as a first step?

Exercise 14: Challenge—reconstruct motion from observations

A localized pulse is measured at successive times. Its maximum is observed at

The pulse shape is unchanged within measurement resolution.

1. Determine the propagation direction.
2. Determine the propagation speed.
3. If the initial profile is F(x − 3), write a translated expression for u(x,t).
4. Predict the position of the maximum at t = 4.0 s.
5. Explain how the measurements support the shape-preserving translation model.

Part II: Complete Worked Solutions

Solution 1: Direction and speed from the translated argument

1. F(x − 4t) has the form F(x − ct), so it moves toward +x with
   
2. G(x + 7t) has the form G(x + ct), so it moves toward −x with
   
3. H(x − 0.25t) moves toward +x with
   
4. The sign is opposite because a fixed feature satisfies
   

   For the minus sign,

   

   so x increases with time. For the plus sign,

   

   so x decreases with time.

Common error. Do not interpret the internal sign as an arrow. Track a feature instead.

Solution 2: Track a fixed feature

A fixed feature satisfies

Therefore

1. The position law is x(t) = 2 + 3t.
2. 
3. 

   The same result follows from Δx = cΔt = (3 m/s)(4 s).

Solution 3: Left-moving feature

The disturbance moves left with speed 5 m/s, so a feature initially at 8 m follows

1. 
2. Set x = −7 m:
   

   Thus

   
3. Direction: −x. Speed:

   5 m/s .

Solution 4: Read pulse translation from snapshots

The pulse center shifts from approximately x = 0 to x = 2 m to x = 4 m.

1. It travels toward increasing x.
2. The center moves
   

   during each one-second interval.

3. 
4. From t = 2 s to 5.5 s,
   

   The center therefore moves

   

   Starting from x = 4 m,

   

Solution 5: Construct translated disturbances

1. Right-moving at 6 m/s:
   
2. Left-moving at the same speed:
   
3. After 2 s each profile has translated a distance
   

   but in opposite directions.

4. No. The function F still describes the same shape. Only the argument changes how the shape is positioned in space.

Solution 6: Recover a snapshot

1. At t = 0,
   
2. At t = 3 s,
   
3. The later profile is the original profile shifted 6 m toward positive x.
4. At t = t₀,
   

Solution 7: Dimensional consistency

For a translated argument, every term combined with x must have dimensions of length.

1. Valid. Since
   

   has units of meters, it can be subtracted from x.

2. Invalid if the number 4 represents a speed. A position cannot be directly subtracted from a speed.
3. Dimensionally valid if the coefficient has units m/s², because
   

   has units of meters. However, this no longer represents translation at constant speed; the shift varies quadratically in time.

4. Valid because vt has dimensions of length.

Key point. Dimensional validity alone does not guarantee the constant-speed WM06 form. Part (c) is dimensionally sensible but is not constant-speed translation.

Solution 8: Gaussian pulse

The pulse is

1. The argument contains x − ct, so the pulse moves toward +x.
2. The exponential is largest when its squared argument is zero:
   

   Therefore

   
3. At t = 3.0 s,
   
4. One width to the right of center means x − ct = a. Hence
   

   Numerically,

   

   With A = 5.0 mm,

   

Solution 9: Spatial snapshot versus time history

1. The left graph is u(x,t₀): a spatial snapshot. The right graph is u(x₀,t): a time history at one fixed observer.
2. At t = 1 s, the pulse center satisfies
   
3. Set the center position equal to the observer location:
   

   Thus

   
4. Both graphs are slices through the same two-variable function u(x,t). One holds t fixed and varies x; the other holds x fixed and varies t.

Solution 10: Propagation versus material motion

1. No. A point on the string does not need to travel horizontally with the pulse.
2. For a transverse string disturbance, the material point can move mainly up and down about its equilibrium position while the disturbance moves along the string.
3. Propagation speed describes how a recognizable feature of the disturbance moves through space. Material-element velocity describes how a particular piece of the medium itself moves.
4. Energy can be transported by the propagating disturbance even though the medium does not undergo bulk transport at the same speed. This distinction becomes essential in the later energy and power articles.

Solution 11: Read an x-t diagram

The slope of a position-versus-time graph is the feature velocity.

1. The line x = 1 + 2t has positive slope and moves toward +x.
2. The line x = 4 − t has negative slope and moves toward −x.
3. Their velocities are
   

   and

   

   respectively.

4. 
5. Set x = 0:
   

   so

   

Solution 12: Diagnose a sign error

1. For a fixed feature,
   

   Therefore

   

   As time increases, x increases.

2. F(x−ct) moves toward

   +x .

3. F(x+ct) moves toward

   −x .

4. Set the argument equal to a constant and solve for x(t). The sign of the resulting slope reveals the direction.

Solution 13: Shape preservation and its limits

1. No, not with one fixed function F. The form F(x−ct) translates an unchanged profile. A widening profile changes shape.
2. Every point of the profile experiences the same horizontal shift ct. Relative separations between features remain unchanged, so the width stays fixed.
3. WM06 mentions effects such as dispersion, attenuation, and reflection. Dispersion in particular can cause different spectral components to travel differently and change the shape.
4. The model isolates propagation kinematics. It allows direction, speed, feature tracking, snapshots, and time histories to be understood before introducing more complicated physics.

Solution 14: Challenge—reconstruct motion from observations

The maximum positions increase by 1.5 m every 0.5 s.

1. Because the maximum moves to larger x, propagation is toward +x.
2. 
3. The initial profile is centered through F(x − 3). Translating that shape rightward at 3 m/s gives
   

   Equivalently, defining a shifted initial-shape function G(x) = F(x − 3) gives u(x,t) = G(x − 3t).

4. The maximum begins at x = 3.0 m and follows
   

   At t = 4.0 s,

   
5. The maximum advances by equal distances in equal times, indicating constant speed, while the statement that the shape is unchanged supports the assumption that one fixed profile is being translated rather than deformed.

What this set prepares you for

WM06E1 has treated translation without assuming a sinusoidal shape. WM07 will combine the sinusoidal spatial pattern developed in WM05 with the translation rule developed in WM06. That produces the standard sinusoidal traveling wave

Because the shape, spatial phase, and translation ideas have already been introduced separately, each part of the WM07 expression will have a physical meaning before the full formula is assembled.

Summary of skills practiced

After completing this set, you should be able to:

• determine propagation direction from F(x ∓ ct);
• identify propagation speed from a translated argument;
• track a fixed feature using x ∓ ct = ξ₀;
• construct right- and left-moving versions of an arbitrary initial shape;
• interpret pulse snapshots and position-time diagrams;
• distinguish a spatial snapshot from a time history;
• distinguish propagation speed from material-element velocity;
• check dimensional consistency of translated arguments;
• recognize the assumption of shape preservation in F(x ∓ ct);
• infer a translation law from measured feature positions.

References

References