Wave Mechanics Examples: Wave Speed

This companion article provides self-study exercises for WM08, wave mechanics: Wave speed. The problems consolidate the equivalent kinematic relations

(1)

and the signed constant-phase velocities for one-dimensional sinusoidal waves. These relations are standard in introductory wave mechanics [1, 2, 3, 4].

All exercises are stated before the worked solutions. The set remains within the WM08 scope: it treats wave speed kinematically and does not yet derive the medium-dependent dynamical value of c from a wave equation.

How to use this problem set

Attempt every exercise in Part I before consulting Part II. For each numerical problem, identify whether the given information is temporal (T,f,ω), spatial (λ,k), or both. For direction questions, distinguish the positive speed magnitude c from the signed phase velocity.

Part I: Exercises

Exercise 1: Frequency and wavelength

A wave has frequency 4.0 Hz and wavelength 1.5 m.

1. Calculate its propagation speed.
2. State the physical meaning of the result in one sentence.
3. If the wavelength doubled while the frequency remained fixed, what would happen to the speed implied by the kinematic relation?

Exercise 2: Angular frequency and angular wavenumber

A right-moving sinusoidal wave has

1. Calculate c from ω∕k.
2. Determine f and λ.
3. Verify the speed independently using fλ.

Exercise 3: Infer wavelength from speed and frequency

A periodic disturbance travels at 150 m∕s and has frequency 60 Hz.

1. Find its wavelength.
2. Find its period.
3. How far does a fixed phase point move during one period?

Exercise 4: Read speed from a constant-phase track

The figure shows the trajectory of one constant-phase feature in an x–t diagram.

1. Read Δx and Δt from the graph.
2. Calculate the signed propagation velocity.
3. State the propagation direction.
4. Give the positive speed magnitude.

Exercise 5: Direction from the wave equation

For each wave, assume k > 0 and ω > 0. Determine the signed phase velocity and propagation direction.

1. u = A cos(kx − ωt).
2. u = A cos(kx + ωt).
3. u = −A cos(kx − ωt + π∕4).
4. Explain why the minus sign in front of A in part (c) does not reverse the propagation direction.

Exercise 6: One wavelength in one period

A crest of a periodic wave is at x = 1.2 m at t = 0.40 s and at x = 3.0 m at t = 1.00 s. Suppose the two events refer to the same crest and are separated by exactly one period.

1. Determine the wavelength.
2. Determine the period.
3. Determine the speed using λ∕T.
4. Determine the frequency and verify the result using fλ.

Exercise 7: Different waves, same speed

The following two spatial profiles belong to waves traveling in the same medium. Their frequencies are f₁ = 1.0 Hz and f₂ = 2.0 Hz.

1. Read λ₁ and λ₂ from the figure.
2. Calculate the speed of each wave.
3. Are the speeds the same?
4. Explain how the frequency and wavelength compensate one another.

Exercise 8: Are the measured quantities self-consistent?

A laboratory report lists

1. Check whether k agrees with the listed wavelength.
2. Check whether ω agrees with the listed frequency.
3. Calculate c using fλ.
4. Calculate c using ω∕k.
5. State whether the four measurements are mutually consistent.

Exercise 9: Wave speed versus material speed

The figure shows a transverse pulse moving along a string while a marked material point on the string moves vertically.

1. Which arrow represents the wave propagation velocity?
2. Which arrow represents a local material velocity?
3. Why are these velocities not generally equal or even parallel?
4. Which velocity is described by c = fλ in this lesson?

Exercise 10: Full wave equation synthesis

Consider

where u and x are in meters and t in seconds.

Determine:

1. the propagation direction;
2. k and ω;
3. λ and f;
4. the wave speed using ω∕k;
5. the wave speed using fλ;
6. whether the two results agree.

Exercise 11: Dimensional checks

Decide whether each proposed expression could represent a speed. Give a brief reason.

1. fλ.
2. f∕λ.
3. ω∕k.
4. k∕ω.
5. λ∕T.

Exercise 12: Constant phase without memorizing the sign rule

A wave has phase

where the coefficient of x is in rad/m and the coefficient of t is in rad/s.

1. Hold 𝜃 constant and solve for x(t).
2. Determine the signed phase velocity.
3. Determine the positive speed magnitude.
4. State the direction of propagation.
5. Explain how this calculation reproduces the usual sign convention.

Exercise 13: A medium with fixed propagation speed

In a particular nondispersive approximation, waves travel at 12 m∕s.

1. Find the wavelength when f = 3.0 Hz.
2. Find the wavelength when f = 6.0 Hz.
3. Find the wavelength when f = 12 Hz.
4. Describe the trend between frequency and wavelength at fixed speed.
5. Does this trend imply that increasing frequency caused the speed to change?

Exercise 14: Capstone—reconstruct the propagation speed

A sinusoidal wave is observed experimentally. Consecutive crests in a spatial snapshot are separated by 0.75 m. At one fixed location, consecutive maxima in the time history are separated by 0.30 s. In an independent measurement, a chosen crest moves 2.50 m in 1.00 s.

1. Determine λ.
2. Determine T and f.
3. Calculate c using λ∕T.
4. Calculate c using fλ.
5. Calculate the crest speed from the independent position measurement.
6. Are all three speed estimates consistent?
7. Calculate k and ω, then verify the same speed using ω∕k.
8. Explain why agreement among all four methods is a strong internal consistency check on the measurements.

Part II: Complete Worked Solutions

Solution 1: Frequency and wavelength

Given

we use

1. 
2. A recognizable phase feature of the wave advances 6.0 m each second.
3. If f remains fixed while λ doubles, then the product fλ doubles. The kinematic relation would therefore imply that c doubles. Whether such a change is physically allowed depends on the system; WM08 itself does not specify the dynamics that set c.

Solution 2: Angular frequency and angular wavenumber

1. 
2. 

   and

   
3. 

   The two methods agree apart from rounding.

Solution 3: Infer wavelength from speed and frequency

1. 
2. 
3. During one period, the same phase feature moves one wavelength, so
   

Solution 4: Read speed from a constant-phase track

From the marked events in the figure,

1. The graph directly gives the changes above.
2. 
3. The positive sign means propagation toward increasing x.
4. The speed magnitude is
   

Solution 5: Direction from the wave equation

1. Constant phase gives kx − ωt = constant, so
   

   The wave moves toward positive x.

2. Constant phase gives kx + ωt = constant, so
   

   The wave moves toward negative x.

3. The internal phase still contains kx − ωt, so
   

   The wave moves toward positive x.

4. Multiplying the displacement by −1 changes crest locations into trough locations, equivalent to a phase offset of π. It does not change the sign of the kx and ωt terms, so the propagation direction is unchanged.

Solution 6: One wavelength in one period

The spatial change of the same crest is

and the elapsed time is

Because the events are separated by one full period,

1. 
2. 
3. 
4. 

   Then

   

Solution 7: Different waves, same speed

From the graph,

1. The wavelength readings are the values above.
2. For wave 1,
   

   For wave 2,

   
3. Yes. The speeds are identical.
4. Doubling the frequency while halving the wavelength leaves the product fλ unchanged.

Solution 8: Are the measured quantities self-consistent?

1. 

   which agrees with the listed k.

2. 

   which agrees with the listed ω.

3. 
4. 
5. Yes. Both the defining relations and the two speed calculations agree, so the four listed measurements are mutually consistent.

Solution 9: Wave speed versus material speed

1. The horizontal arrow following the pulse represents the wave propagation velocity.
2. The vertical arrow at the marked string element represents a local material velocity.
3. A wave is a propagating pattern. The medium can oscillate locally while the pattern moves through it, so the two velocities need not have the same magnitude or direction.
4. The relation c = fλ describes the propagation speed of the wave pattern or a constant-phase feature.

Solution 10: Full wave equation synthesis

The wave is

1. The phase contains kx − ωt, so the wave propagates toward positive x.
2. 
3. 

   and

   
4. 
5. 
6. Yes. The two forms give the same result.

Solution 11: Dimensional checks

1. fλ has units
   

   so it can represent a speed.

2. f∕λ has units 1∕(m s), not speed.
3. ω∕k has units
   

   so it can represent a speed.

4. k∕ω has units s∕m, the reciprocal of speed.
5. λ∕T has units m∕s, so it can represent a speed.

Solution 12: Constant phase without memorizing the sign rule

Set

Then

so

1. The expression above is the constant-phase trajectory.
2. Its slope is
   
3. 
4. The negative sign means motion toward decreasing x.
5. Because the phase contains kx + ωt, solving for constant phase necessarily produces a negative slope. This is the origin of the usual sign rule.

Solution 13: A medium with fixed propagation speed

At fixed c,

1. 
2. 
3. 
4. At fixed speed, wavelength is inversely proportional to frequency. Doubling f halves λ.
5. No. In this exercise the speed was stipulated to remain fixed. The changing wavelength is what allows fλ to remain constant.

Solution 14: Capstone—reconstruct the propagation speed

The spatial data give

The temporal data give

1. 
2. 
3. 
4. 
5. The crest-position measurement gives
   
6. Yes. The three methods agree.
7. 

   and

   

   Therefore

   
8. Agreement shows that the measured spatial period, temporal period, crest displacement, and angular quantities all describe the same translating wave. A disagreement would signal either measurement error or an incorrect model assumption.

What this set completes

WM08E1 closes the first self-study block of the Wave Mechanics series. After WM00–WM08 and their companions, the student should be able to move freely between

and interpret the sinusoidal traveling-wave phase

The next block begins with WM09 and develops the two-variable field u(x,t) more formally, leading eventually to partial derivatives and the one-dimensional wave equation.

References

References