Wave Mechanics Examples: The Sinusoidal Traveling Wave

This companion article provides self-study exercises for WM07, wave mechanics: The Sinusoidal Traveling Wave. The central one-dimensional forms are

(1)

for propagation toward increasing x, and

(2)

for propagation toward decreasing x, when k > 0 and ω > 0. These forms are standard in introductory wave mechanics [1, 2, 3, 4].

The exercises stay within the WM07 scope. They develop interpretation of the complete phase, extraction of wavelength and period, evaluation of a wave at a specific event (x,t), fixed-time spatial snapshots, fixed-position time histories, phase offsets, and propagation direction. The explicit wave-speed relations c = ω∕k = fλ are reserved for WM08.

How to use this problem set

Attempt every exercise in Part I before consulting Part II. When a propagation direction is requested, identify the sign of the temporal term or verify the answer by following a constant phase. For numerical work, keep the distinction between spatial quantities (k,λ) and temporal quantities (ω,T,f) visible at every step.

Part I: Exercises

Exercise 1: Read the parameters from the equation

Consider

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

Determine:

1. the amplitude A;
2. the Wavenumber k;
3. the angular frequency ω;
4. the phase constant ϕ;
5. the propagation direction;
6. the wavelength λ;
7. the period T and ordinary frequency f.

Exercise 2: Right-moving or left-moving?

Assume every listed k and ω is positive. State the propagation direction of each disturbance.

1. u₁ = A cos(kx − ωt).
2. u₂ = A cos(kx + ωt).
3. u₃ = A sin(kx − ωt + π∕3).
4. u₄ = −A cos(kx + ωt).
5. Explain why the minus sign in front of A in part (d) does not reverse the propagation direction.

Exercise 3: Construct the wave equation

A sinusoidal wave has amplitude 3.0 cm, wavelength 0.80 m, period 0.50 s, and phase constant π∕6. It propagates toward positive x.

1. Calculate k.
2. Calculate ω.
3. Write the wave function in the form A cos(kx−ωt+ϕ), with u expressed in centimeters.
4. Write the corresponding form if the same sinusoidal pattern instead propagates toward negative x.

Exercise 4: Read a traveling wave from snapshots

The figure shows the same sinusoidal wave at two instants separated by 0.25 s.

Determine:

1. the amplitude;
2. the wavelength;
3. the direction of propagation;
4. the distance translated during the time interval;
5. what fraction of one wavelength that translation represents;
6. the period implied by the two snapshots.

Exercise 5: Evaluate the wave at one event

Let

where u is in millimeters, x in meters, and t in seconds.

Find:

1. the phase 𝜃 at x = 0.50 m and t = 0.125 s;
2. the displacement at that event;
3. whether the result lies within the physically allowed range of the wave.

Exercise 6: Extract a spatial snapshot

A right-moving wave is

with u in centimeters, x in meters, and t in seconds.

1. Write the spatial snapshot u(x, 0).
2. Write the spatial snapshot at t = 1∕6 s.
3. Determine the wavelength of either snapshot.
4. Explain why changing the snapshot time changes the phase offset but not the wavelength.

Exercise 7: Extract a time history

For the same wave used in Exercise 6,

find the time history observed at x = 0.50 m.

1. Simplify u(0.50,t) as far as practical.
2. Determine the period.
3. Determine the frequency.
4. Explain why the time history contains the same ω as the full traveling-wave expression.

Exercise 8: Spatial snapshot versus time history

The following two graphs are generated from the same traveling sinusoid.

1. Which horizontal axis represents position and which represents time?
2. Read the wavelength from the spatial plot.
3. Read the period from the temporal plot.
4. Calculate k and ω.
5. Explain why both graphs can describe the same two-variable function u(x,t).

Exercise 9: The role of the phase constant

Consider the two right-moving waves

and

1. What is the phase difference Δϕ = ϕ₂ − ϕ₁?
2. At t = 0, how far in x is a corresponding crest of u₂ shifted relative to a crest of u₁? Express the answer in terms of λ.
3. Does the phase constant change the wavelength?
4. Does the phase constant reverse the propagation direction?

Exercise 10: Sine and cosine representations

Rewrite each wave in an equivalent form using the other trigonometric function.

1. u = A cos(kx − ωt) as a sine.
2. u = A sin(kx − ωt) as a cosine.
3. Explain why the replacement does not change the physical class of wave represented.

Exercise 11: Dimensional consistency of complete phase

Assume x is measured in meters and t in seconds. For each expression, decide whether its trigonometric argument is dimensionally valid.

1. cos[(4 m⁻¹)x − (7 s⁻¹)t].
2. cos[(4 m⁻¹)x − 7t] when the number 7 is dimensionless.
3. cos[kx − ωt + ϕ] with k in rad/m, ω in rad/s, and ϕ in radians.
4. cos(k − ωt).

Exercise 12: Verify direction using constant phase

For the wave

a marked crest corresponds to a fixed phase 𝜃₀.

1. Write the constant-phase equation.
2. Solve that equation for x as a function of t.
3. Without introducing a numerical wave speed, explain from the sign of the time-dependent term why the marked crest moves toward increasing x.
4. Repeat the reasoning for A cos(kx + ωt + ϕ).

Exercise 13: Read a constant-phase track

The figure shows one constant-phase feature of a sinusoidal wave in an x–t diagram.

1. Is the wave propagating toward +x or −x?
2. Which of the two phase forms, kx−ωt + ϕ or kx + ωt + ϕ, is consistent with the track for positive k and ω?
3. What does it mean physically that every point on the line has the same phase value?
4. What additional relation will WM08 obtain from the slope of this line?

Exercise 14: Synthesis from measured wave data

A sinusoidal disturbance has the following measured properties:

• maximum displacement magnitude A = 12 mm;
• adjacent crests are separated by 0.60 m;
• at a fixed detector the time between consecutive maxima is 0.24 s;
• the wave moves toward positive x;
• at x = 0 and t = 0, the wave is at its positive maximum.

1. Find k.
2. Find ω.
3. Choose a convenient phase constant ϕ.
4. Construct a cosine traveling-wave function consistent with all of the observations.
5. Verify from your equation that u(0, 0) = +A.
6. State which measured fact determines the sign of the temporal term.

Part II: Complete Worked Solutions

Solution 1: Read the parameters from the equation

Compare

with

1. 
2. 
3. 
4. 
5. The phase has the form kx−ωt+ϕ, so for positive k and ω the wave propagates toward
   

6. λ	=	(3)
   	=	(4)
   	= 0.40 m .	(5)

7. T	= = = 0.25 s ,	(6)
   f	= = 4.0 Hz .	(7)

Common error. The coefficient 5π is k, not the wavelength. Wavelength is obtained from 2π∕k.

Solution 2: Right-moving or left-moving?

1. kx − ωt corresponds to
   
2. kx + ωt corresponds to
   
3. Replacing cosine with sine changes only the phase convention. The kx − ωt structure still gives
   
4. The phase still contains kx + ωt, so the wave travels toward
   
5. The factor −A flips the disturbance vertically. It is equivalent to adding a phase shift of π because
   

   It does not change the sign that controls translation through space.

Solution 3: Construct the wave equation

The data are

1. k	=	(8)
   	=	(9)
   	= 2.5π rad/m .	(10)

2. ω	=	(11)
   	=	(12)
   	= 4π rad/s .	(13)

3. Positive-x propagation uses the minus temporal sign:
   
4. For negative-x propagation,
   

Solution 4: Read a traveling wave from snapshots

From the plotted vertical scale, the extrema are +2 cm and −2 cm.

1. Therefore
   
2. Consecutive crests in either snapshot are separated by
   
3. The marked crest shifts from x = 1 m to x = 2 m as time increases, so propagation is toward
   
4. The translation is
   
5. Relative to the wavelength,
   
6. A shift of λ∕4 corresponds to one quarter of a cycle in time. Since the snapshots are separated by 0.25 s,
   

   and hence

   

Solution 5: Evaluate the wave at one event

The phase is

At x = 0.50 m and t = 0.125 s,

𝜃	= 2π(0.50) − 4π(0.125) +	(14)
	= π − +	(15)
	= .	(16)

Thus

u	= 6.0 cos mm	(17)
	= 6.0 mm	(18)
	= −3 mm	(19)
	≈−5.20 mm .	(20)

Since A = 6.0 mm, the allowed range is

The result lies inside this range, so it is physically consistent.

Solution 6: Extract a spatial snapshot

The full wave is

1. At t = 0,
   
2. At t = 1∕6 s,
   

   so

   u	= 2.5 cos(3πx − π + π∕4) cm	(21)
   	= 2.5 cos(3πx − 3π∕4) cm .	(22)

3. Since k = 3π rad/m,
   
4. Changing t changes only the additive phase offset −ωt + ϕ. The coefficient of x remains k = 3π, so the wavelength remains unchanged.

Solution 7: Extract a time history

At x = 0.50 m,

u(0.50,t)	= 2.5 cos	(23)
	= 2.5 cos	(24)
	= 2.5 cos cm .	(25)

The angular frequency is still

Therefore

T	= = s ,	(26)
f	= = 3.0 Hz .	(27)

Fixing x replaces kx by a constant phase contribution. It does not alter the coefficient of t, so the time history retains the same ω as the full traveling wave.

Solution 8: Spatial snapshot versus time history

1. The left graph has horizontal axis x and is the spatial snapshot. The right graph has horizontal axis t and is the time history.
2. The marked crest-to-crest separation is
   
3. The marked maximum-to-maximum time separation is
   

4. k	= = = π rad/m ,	(28)
   ω	= = = 4π rad/s .	(29)

5. A two-variable wave function contains both dependences. Setting t = t₀ produces a function of x only; setting x = x₀ produces a function of t only. These are different slices through the same u(x,t).

Solution 9: The role of the phase constant

1. 
2. At t = 0, a crest satisfies a constant phase. For u₁ choose kx = 0. For u₂, a corresponding crest can satisfy
   

   so

   

   Using k = 2π∕λ,

   

   Thus, at the same reference time, the u₂ crest is shifted one quarter wavelength toward smaller x relative to the corresponding u₁ crest.

3. No. The coefficient k is unchanged, so λ = 2π∕k is unchanged.
4. No. Both phases contain kx − ωt, so both waves propagate toward +x.

Solution 10: Sine and cosine representations

Use

and

1. 
2. 
3. The change merely alters the phase constant. Wavelength, period, amplitude, and propagation direction remain the same. Thus sine and cosine are two phase conventions for the same class of sinusoidal traveling wave [3, 1].

Solution 11: Dimensional consistency of complete phase

1. Valid. The products
   

   are dimensionless angular quantities.

2. Invalid as written. If 7 is dimensionless, then 7t has units of time and cannot be subtracted from a dimensionless spatial phase.
3. Valid. kx, ωt, and ϕ are all angular phase terms.
4. Invalid. k alone has dimensions of inverse length, whereas ωt is dimensionless. The spatial term must be multiplied by a length such as x.

Solution 12: Verify direction using constant phase

A fixed feature satisfies

Solving for x gives

Since k > 0 and ω > 0, increasing t requires increasing x. Therefore the feature moves toward +x.

For the opposite sign,

so

As t increases, x decreases. Therefore the feature moves toward −x.

The appearance of the ratio ω∕k anticipates WM08, where that ratio is identified and interpreted explicitly as the propagation speed.

Solution 13: Read a constant-phase track

1. The feature appears at larger x at later t, so the wave moves toward
   
2. For positive k and ω, this is consistent with
   
3. Every point on the line represents a different event (x,t) at which the wave has the same phase. Thus the plotted line follows one recognizable wave feature such as a crest.
4. WM08 will use the slope of the constant-phase line to obtain the propagation speed and show that
   

Solution 14: Synthesis from measured wave data

The measured quantities are

1. k	=	(30)
   	= rad/m .	(31)

2. ω	=	(32)
   	= rad/s .	(33)

3. At x = 0 and t = 0 the wave is at its positive maximum. For a cosine representation, the simplest choice is
   

   Any phase equivalent to 2πn would describe the same starting state.

4. Since the wave moves toward positive x, use the minus temporal sign:
   
5. At the origin of space and time,
   
6. The observation that the wave moves toward positive x determines the minus sign in kx − ωt.

What this set prepares you for

WM07E1 has treated the sinusoidal traveling wave as a kinematic object. The student can now identify every parameter in

switch between spatial snapshots and time histories, evaluate the disturbance at an event, and follow constant phase through space and time. WM08 will now make the natural quantitative connection between the spatial and temporal rates of phase change by deriving the wave speed.

Summary of skills practiced

After completing this set, you should be able to:

• read A, k, ω, and ϕ from a traveling-wave equation;
• determine propagation direction from the phase sign convention;
• recover λ, T, and f from k and ω;
• construct a sinusoidal traveling-wave equation from physical data;
• evaluate u(x,t) at a specified event;
• extract a fixed-time spatial snapshot and a fixed-position time history;
• interpret the phase constant as a shift rather than a change of direction;
• convert between sine and cosine phase conventions;
• check dimensional consistency of a complete wave phase;
• track a constant-phase feature in an x–t diagram.

References

The notation and traveling-wave conventions used in this problem set follow the same sources used in WM07.

References