where we have assumed that the amplitude is zero at the origin, that is for . Recall both and have dimension of length. We need quantities and in eq.(5.1) from dimensional analysis. Since the argument of a sine function is dimensionless, we need a parameter which has dimension inverse of length, that is ; furthermore, since the sine function is dimensionless, we need a quantity with dimension to make eq.(5.1) consistent. What is the physical significance of the quantities and ? The maximum value of the sine function is 1, and hence from eq.(5.1) we see that , called the amplitude of the wave, is the maximum height that an string can have, and is shown in Figure 5.2. To understand what is , called the wave number, note since the sine function is periodic, we have

(5.2) | |||

(5.3) |

From the above, we see that if is increased by an amount the pattern of the wave

(5.4) | |||

(5.5) | |||

(5.6) |

What is the shape of the string at some later time ? We consider only the simplest case for which the entire wave is traveling at a constant velocity of ; in other words

(5.7) |

As the wave propagates, we see from the example above, that we have to increase the value of m by an amount of () m, which is the distance covered by the wave in time s. To understand the general case,let us follow the motion of a point of the wave which is at at time , and whose value is given by

(5.9) |

(5.10) | |||

(5.11) | |||

(5.12) |

We see from the above equation that the position and time coordinates of the wave only appears in the combination . Hence in general, the shape of the wave for time and at position is given by

One can easily see that we recover the special case given in eq.(5.8) from eq.(5.13). We hence have the result that the wave is propagating in the -direction with a

(5.15) | |||

(5.16) |

We see that the material of the medium at point does not travel in the -direction, but rather, oscillates back and forth in the -direction. An important characteristic of a traveling wave is that the medium oscillates in the transverse -direction with the

The result we have obtained in eq.(5.19) has a simple interpretation. Frequency refers to how rapidly the material points of the medium are oscillating about their equilibrium position. Eq.(5.19) tells us that in the time that it takes for a point of the wave moves through the distance of one full wavelength, namely , the material particle, in time interval , undergoes one complete oscillation.

Collecting the results for the wave , we obtain the standard
expression for a wave that is given by

Note from the above that the wave at the point undergoes motion in the -direction - which is perpendicular to the direction of propagation, as is expected of a

We have derived the important result that the amount of

(5.25) |

The

where we have used eq(5.28) to obtain the last line. The most important fact about intensity is that it is proportional to ; in other words, intensity is proportional to .

*Energy Propagation in Waves

Recall that the phase is fixed by (3.91), and is determined by the position that the particle occupies at say . To create the simple wave motion (harmonic) in the string, we have to start all the different particles

If we plot the positions of the various particles at a given instant as shown in the Figure 5.8, we see that the particles all start their oscillations from an initial shape of a wave. Hence, for let , the positions of the particles are then given, from eqns.(5.32), by

The transverse velocity is given, as in the continuous case of eq.(5.20), by

By comparing with eq.(5.13) we see that the wave is already looking a lot like the motion of a number of discrete particles. The energy of this system of -particles undergoing harmonic oscillations is simply the sum of the energies of the individual particles, denoted by . That is, energy of wave is

(5.34) | |||

(5.35) |

For a

where is the tension of the string. From eqs.(5.34) and (5.35), we have the important result

In other words, each elemental mass at position has a constant energy, as is expected from our earlier discussion on simple harmonic motion. Note that the energy being propagated is proportional to the

(5.40) | |||

(5.41) |

Hence, the power expended in sustaining wave propagation is given by

(5.42) | |||

(5.43) |

where is the velocity of the wave. Note we have derived the equation above in eq. (wpwr2) using much simpler graphical means. In Figure 5.4 we have seen that the power that is needed to create a propagating wave goes into creating a

From dimensional analysis, we can straight away conclude that the
velocity of propagation for a taut string
; a more careful analysis from
eq.(5.48) shows that

The resultant wave is said to be the result of the interference of the two waves. The resulting wave has an intensity, from eq.(5.31), given by

Note the above equation is a characteristic feature of wave behaviour, and will be of great significance in our discussion on quantum mechanics. The generelization to the superposition of arbitrary number waves is straightforward. For simplicity, consider the case of two waves having the same amplitude and frequency, but which start off with two different initial shapes. Let the first wave be denoted by

(5.49) | |||

(5.50) | |||

(5.51) |

and the second wave be denoted by

(5.52) |

From the rules of trigonometry, the sum of two sines is given by

(5.55) |

In general, the resultant wave is a traveling wave, as can be seen from its dependence on There are two notable features of . (a) The intensity of the resultant is shown in Figure 5.10 wave, and is proportional to , and depends on the phase and is given by phase difference, namely, is a function of (b)The phase of the resultant wave, , is given by . Two special cases for the phase difference are given below.

In eq.(5.65) above the component waves are said to

Note from eq.(5.71) that at the points , called the nodes of the standing wave, the medium is fixed and hence

Resonance

But we are not done; we have to account for the fact that the two ends of the string are always fixed. Suppose the length of the violin is , with one fixed end at and the other end at . We then must have

Hence, unlike the case for an arbitrary standing waves, the violin string can have only wavelengths which are equal to a multiple of as given in eq.(5.75). The reason for this result is easy to understand. Only waves that have wavelengths that are integer multiples of can fit into the length of the violin as standing waves. All other wavelengths, as was the case of a standing wave, are eliminated by destructive interference. Furthermore, since wavelength is related to frequency by eq.(), we have from eq.(5.49)

(5.71) |

A particular oscillation of the bangle (ring) is shown in Figure 5.14.

(5.74) | |||

(5.75) | |||

(5.76) |

To understand the physical phenomenon that the equation above refers to, suppose ; we then have

(5.77) |

Doppler Effect

Suppose the source is moving at a speed ; then, after emitting the first signal, it travels a distance of , and then emits the second signal. Hence, as shown in Figure 5.15 the distance between the first signal and the second is now a

(5.80) | |||

(5.81) | |||

(5.82) | |||

(5.83) |

Hence, the frequency of sound from a moving source, and received by a stationary observer, is

Clearly, since , we have from eq.(5.91) that ; in other words the frequency is of higher pitch, as expected for a source of sound that is

(5.87) |

(5.88) |