# 1.4 Traveling Waves

Waves are a natural consequence of many physical processes: Waves manifest themselves as ripples on the surfaces of oceans and lakes; sound waves constitute pressure disturbances that travel through air; mechanical waves modulate stretched strings; and electromagnetic waves carry electric and magnetic fields through free space and material media as microwaves, light, and X-rays. All of these various types of waves exhibit a number of common properties, including:

- Moving waves carry energy.

- Waves have velocity; it takes time for a wave to travel from one point to another. Electromagnetic waves in a vacuum travel at a speed of 3×10 8 m/s, and sound waves in air travel at a speed approximately a million times slower, specifically 330 m/s. Sound waves cannot travel in a vacuum.

- Many waves exhibit a property called linearity. Waves that do not affect the passage of other waves are called linear because they can pass right through each other. The total of two linear waves is simply the sum of the two waves as they would exist separately. Electromagnetic waves are linear, as are sound waves. When two people speak to one another, the sound waves they generate do not interact with one another but simply pass through each other. Water waves are approximately linear; the expanding circles of ripples caused by two pebbles thrown into two locations on a lake’s surface do not affect each other. Although the interaction of the two circles may exhibit a complicated pattern, it is simply the linear superposition of two independent expanding circles.

Waves are of two types: transient waves caused by sudden disturbances and continuous periodic waves generated by a repetitive source. We encounter both types of waves in this book, but most of our discussion deals with the propagation of continuous waves that vary sinusoidally with time.

An essential feature of a propagating wave is that it is a self-sustaining disturbance of the medium through which it travels. If this disturbance varies as a function of one space variable, such as the vertical displacement of the string shown in Fig. 1-10, we call the wave one-dimensional. The vertical displacement varies with time and with the location along the length of the string. Even though the string rises up into a second dimension, the wave is only one-dimensional because the disturbance varies with only one space variable. A two-dimensional wave propagates out across a surface, like the ripples on a pond (Fig. 1-11(a)), and its disturbance can be described by two space variables. By extension, a three-dimensional wave propagates through a volume, and its disturbance may be a function of all three space variables. Three-dimensional waves may take on many different shapes; they include plane waves, cylindrical waves, and spherical waves. A plane wave is characterized by a disturbance that at a given point in time has uniform properties across an infinite plane perpendicular to its direction of propagation (Fig. 1-11(b)). Similarly, for cylindrical and spherical waves, the disturbances are uniform across cylindrical and spherical surfaces (Figs. 1-11(b) and (c)).

# Figure 1-10 A one-dimensional wave traveling on a string.

Figure 1 10 Full Alternative Text

# Figure 1-11 Examples of two-dimensional and three-dimensional waves: (a) circular waves on a pond, (b) a plane light wave exciting a cylindrical light wave through the use of a long narrow slit in an opaque screen, and (c) a sliced section of a spherical wave.

(a) Circular waves

(b) Plane and cylindrical waves

Figure 1 11-b Full Alternative Text

(c) Spherical wave

In the material that follows, we examine some of the basic properties of waves by developing mathematical formulations that describe their functional dependence on time and space variables. To keep the presentation simple, we limit our discussion to sinusoidally varying waves whose disturbances are functions of only one space variable, and we defer the discussion of more complicated waves to later chapters.

# 1.4.1 Sinusoidal Waves in a Lossless Medium

Regardless of the mechanism responsible for generating them, all linear waves can be described mathematically in common terms.

By way of an example, let us consider a wave traveling on a lake’s surface, and let us assume for the time being that frictional forces can be ignored, thereby allowing a wave generated on the water’s surface to travel indefinitely with no loss in energy. If y denotes the height of the water’s surface relative to the mean height (undisturbed condition) and x denotes the distance of wave travel, the functional dependence of y on time t and the spatial coordinate x has the general form

where A is the amplitude of the wave, T is its time period, λ is its spatial wavelength, and ϕ 0 is a reference phase. The quantity y(x,t) also can be expressed in the form

y(x,t)=A cos ϕ(x,t)(m),(1.18)

where

ϕ(x,t)=2 π t T−2 π x λ+ϕ 0(rad).(1.19)

The angle ϕ(x,t) is called the phase of the wave, and it should not be confused with the reference phase ϕ 0, which is constant with respect to both time and space. Phase is measured by the same units as angles, that is, radians (rad) or degrees, with 2 π radians=360°.

Let us first analyze the simple case when ϕ 0=0:

y(x,t)=A cos 2 π t T−2 π x λ(m). (1.20)

The plots in Fig. 1-12 show the variation of y(x,t) with x at t=0 and with t at x=0. The wave pattern repeats itself at a spatial period λ along x and at a temporal period T along t.

# Figure 1-12 Plots of y(x,t)=A cos 2 π t T−2 π x λ as a function of (a) x at t=0 and (b) t at x=0.

(a) y(x,t) versus x at t = 0

Figure 1 12-a Full Alternative Text

(b) y(x,t) versus t at x = 0

Figure 1 12-b Full Alternative Text

If we take time snapshots of the water’s surface, the height profile y(x,t) would exhibit the sinusoidal patterns shown in Fig. 1-13. All three profiles correspond to three different values of t, and the spacing between peaks is equal to the wavelength λ, even though the patterns are shifted relative to one another because they correspond to different observation times. Because the pattern advances along the +x direction at progressively increasing values of t, y(x,t) is called a wave traveling in the +x direction. If we track a given point on the wave, such as the peak P, and follow it in time, we can measure the phase velocity of the wave. At the peaks of the wave pattern, the phase ϕ(x,t) is equal to zero or multiples of 2 π radians. Thus,

ϕ(x,t)=2 π t T−2 π x λ=2 n π,n=0,1,2, ...(1.21)

Had we chosen any other fixed height of the wave, say y 0, and monitored its movement as a function of t and x, this again would have been equivalent to setting the phase ϕ(x,t) constant such that

y(x,t)=y 0=A cos 2 π t T−2 π x λ(1.22)

or

2 π t T−2 π x λ=cos−1 y 0 A=constant.(1.23)

The apparent velocity of that fixed height is obtained by taking the time derivative of Eq. (1.23), so

2 π T−2 π λ d x d t=0,(1.24)

which gives the phase velocity u p as

The water itself mostly moves up and down; when the wave moves from one point to another, the water does not move physically along with it.

# Figure 1-13 Plots of y(x,t)=A cos 2 π t T−2 π x λ as a function of x at (a) t=0, (b) t=T/4, and (c) t=T/2. Note that the wave moves in the +x direction with a velocity u p=λ/T.

Figure 1 13 Full Alternative Text

The frequency of a sinusoidal wave, f, is the reciprocal of its time period T:

Combining the preceding two equations yields

The wave frequency f is measured in cycles per second and has been assigned the unit hertz (Hz), named in honor of the German physicist Heinrich Hertz (1857–1894), who pioneered the development of radio-wave instrumentation.

Using Eq. (1.26), Eq. (1.20) can be rewritten in a more compact form as

where ω is the angular velocity of the wave and β is its phase constant (or wavenumber), defined as

In terms of these two quantities,

u p=f λ=ω β.(1.30)

So far, we have examined the behavior of a wave traveling in the +x direction. To describe a wave traveling in the −x direction, we reverse the sign of x in Eq. (1.28):

We now examine the role of the phase reference ϕ 0 given previously in Eq. (1.17). If ϕ 0 is not zero, then Eq. (1.28) should be written as

y(x,t)=A cos(ω t−β x+ϕ 0). (1.32)

A plot of y(x,t) as a function of x at a specified t or as a function of t at a specified x is shifted in space or time, respectively, relative to a plot with ϕ 0=0 by an amount proportional to ϕ 0. This is illustrated by the plots shown in Fig. 1-14. We observe that when ϕ 0 is positive, y(t) reaches its peak value, or any other specified value, sooner than when ϕ 0=0. Thus, the wave with ϕ 0=π/4 is said to lead the wave with ϕ 0=0 by a phase lead of π/4; and similarly, the wave with ϕ 0=−π/4 is said to lag the wave with ϕ 0=0 by a phase lag of π/4. A wave function with a negative ϕ 0 takes longer to reach a given value of y(t), such as its peak, than the zero-phase reference function.

# Figure 1-14 Plots of y(0,t)=A cos[(2 π t/T)+ϕ 0] for three different values of the reference phase ϕ 0.

Figure 1 14 Full Alternative Text

# 1.4.2 Sinusoidal Waves in a Lossy Medium

If a wave is traveling in the x direction in a lossy medium, its amplitude decreases as e−α x. This factor is called the attenuation factor, and α is called the attenuation constant of the medium and its unit is neper per meter (NP/m). Thus, in general,

y(x,t)=A e−α x cos(ω t−β x+ϕ 0).(1.33)

The wave amplitude is now A e−α x, not just A. Figure 1-15 shows a plot of y(x,t) as a function of x at t=0 for A=10 m, λ=2 m, α=0.2 Np/m, and ϕ 0=0. Note that the envelope of the wave pattern decreases as e−α x.

# Figure 1-15 Plot of y(x)=(10 e−0.2 x cos π x) meters. Note that the envelope is bounded between the curve given by 10 e−0.2 x and its mirror image.

Figure 1 15 Full Alternative Text

The real unit of α is (1/m); the neper (Np) part is a dimensionless, artificial adjective traditionally used as a reminder that the unit (Np/m) refers to the attenuation constant of the medium, α. A similar practice is applied to the phase constant β by assigning it the unit (rad/m) instead of just (1/m).

Concept Question 1-6: How can you tell if a wave is traveling in the positive x direction or the negative x direction?

Concept Question 1-7: How does the envelope of the wave pattern vary with distance in (a) a lossless medium and (b) a lossy medium?

Concept Question 1-8: Why does a negative value of ϕ 0 signify a phase lag?

# Example 1-1: Sound Wave in Water

An acoustic wave traveling in the x direction in a fluid (liquid or gas) is characterized by a differential pressure p(x,t). The unit for pressure is newton per square meter (N/m 2). Find an expression for p(x,t) for a sinusoidal sound wave traveling in the positive x direction in water given that the wave frequency is 1 kHz, the velocity of sound in water is 1.5 km/s, the wave amplitude is 10 N/m 2, and p(x,t) was observed to be at its maximum value at t=0 and x=0.25 m. Treat water as a lossless medium.

Solution: According to the general form given by Eq. (1.17) for a wave traveling in the positive x direction,

p(x,t)=A cos 2 π T t−2 π λ x+ϕ 0(N/m 2).

The amplitude A=10 N/m 2, T=1/f=10−3 s, and from u p=f λ,

λ=u p f=1.5×10 3 10 3=1.5 m.

Hence,

p(x,t)=10 cos 2 π×10 3 t−4 π 3 x+ϕ 0 (N/m 2).

Since at t=0 and x=0.25 m, p(0.25,0)=10(N/m 2), we have

10=10 cos−4 π 3 0.25+ϕ 0=10 cos−π 3+ϕ 0,

which yields the result (ϕ 0−π/3)=cos−1(1), or ϕ 0=π/3. Hence,

p(x,t)=10 cos 2 π×10 3 t−4 π 3 x+π 3 (N/m 2).

# Example 1-2: Power Loss

A laser beam of light propagating through the atmosphere is characterized by an electric field given by

E(x,t)=150 e−0.03 x cos(3×10 15 t−10 7 x)(V/m),

where x is the distance from the source in meters. The attenuation is due to absorption by atmospheric gases. Determine

1. the direction of wave travel,

2. the wave velocity, and

3. the wave amplitude at a distance of 200 m.

Solution: (a) Since the coefficients of t and x in the argument of the cosine function have opposite signs, the wave must be traveling in the +x direction.

(b)

u p=ω β=3×10 15 10 7=3×10 8 m/s,

and this is equal to c, which is the velocity of light in free space.

(c) At x=200 m, the amplitude of E(x,t) is

150 e−0.03×200=0.37(V/m).