9.2: Progressive Waves
- Page ID
- 45580
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Waves that we create by throwing a rock into a lake and waves that we see on the ocean move freely on the water surface. Therefore, they are called progressive waves (Fig. 9-2). Another type of wave, called a standing wave, behaves differently, as discussed later in this chapter.
If we look carefully at the ripples caused by tossing a rock into a lake, we see that a series of waves, not just a single wave, moves out from the impact point. Each of the individual waves has a rounded top where the water surface is elevated (Fig. 9-2a). The point of highest elevation of the wave is the wave crest. Between each successive pair of crests is a rounded depression of the water surface, or trough. Most progressive waves, whether in a lake or in the deep ocean, also have this general shape.
Waves also have several other important characteristics (Fig. 9-2a). The distance between two adjacent crests (or troughs) is called the wavelength, represented as L. Wave height (H) is the vertical distance between crest and trough, and wave amplitude is equal to H/2, the vertical distance between the crest or trough and the mean water level. Wave period (T) is the time the wave takes to move a distance equal to one wavelength. It is equivalent to the time that elapses between the arrival of two successive crests (or troughs) at a point on the surface. Wave period is measured as the number of seconds per wave. Wave frequency (f) is the number of wave crests (or troughs) that pass a point on the ocean surface in a given time. It is measured as the number of waves (or fractions of a wave) per second. Wave period (T), usually measured in seconds, and wave frequency (f) are related by a simple equation:
T = 1/f
or
f = 1/T
We can estimate frequency without a stopwatch by counting the number of waves that pass in several minutes and dividing by the elapsed time (in seconds).
Two other important characteristics of progressive waves, wave speed and wave steepness, can be determined from the wavelength, wave height, and period or frequency. Wave speed is often called “celerity” (abbreviated C) because there is essentially no net forward movement (or speed) of the water as the wave passes. As explained later in this chapter, only the wave energy and waveform, and not the water, move forward with the wave. Wave speed (C) is the ratio of the wavelength to the wave’s period and can be calculated using the following equations:
C = L/T
or
C = f × L
Wave steepness is equal to the ratio of wave height to wavelength (H/L).