Hakim SAIBI November 4th , 2015
Waves Lecture No. 5 Hakim SAIBI November 4th , 2015 Outline Periodic waves. Waves in three dimensions Harmonic waves.
Energy transfer via waves on a string. Harmonic sound waves. Energy of sound waves. Electromagnetic waves. Waves in three dimensions Wave intensity Intensity level Fig.1. Harmonic wave at some instant in time.
Periodic waves Harmonic waves As the wave propagates along the string move direction of propagation-in simple harmonic motion with the frequency f of the tuning fork. During one period T of this motion the wave moves a distance of one wavelength, so its speed is given by: The sine function that describes the displacements in Fig.1 is: (1) Fig.1. Harmonic wave at some instant in time. (2) Periodic waves Where A is the amplitude, is the wavelength, and is a phase constant that depends on the choice of the origin (where x=0). This equation is expressed more simply as: Where k, called the wave number, is given by: Note that k has units of m-1. (Because the angle must be in radians, we sometimes write the units of k as rad/m) When dealing with a single harmonic wave we usually choose the location of the origin so that =0. For a wave traveling in the direction of increasing x with speed v, replace x in Eq.3 with x-vt. With equal to zero, this gives: or Where Is the angular frequency, and the argument of the sine function, (kx-vt), is called the phase. (3) (4) (5) Harmonic Wave Function (6) Periodic waves The angular frequency is related to the frequency f and period T by: Substituting =2f into Eq.6 and using k=2/, we obtain: Or v=f, which is Eq.1. If a harmonic wave traveling along a string is described by y(x,t)=Asin(kx-t), the velocity of a point on the string at a fixed value of x is: The acceleration of this point is given by 2y/ t2. (7) (8) Transverse Velocity Periodic waves Energy transfer via waves on a string
Consider again a string attached to a tuning fork. As the fork vibrates, it transfers energy to the segment of the string attached to it. For example, as the fork moves upward from its equilibrium position it stretches the adjacent string segment slightly-increasing its elastic potential energy. In addition, the fork slows as it moves upward from it equilibrium, so it slows the string segment closest to it. This decreases the kinetic energy of the segment. As a wave moves along the string, energy is transferred from one segment to the next in a similar manner. Power is the rate of energy transfer. We can calculate the power by considering work done by the force that one segment of the string exerts on a neighboring segment. The rate of work done by this force is the power. Fig.2 shows a harmonic wave moving to the right along a string segment. That is, we assume a wave function of the form: (9) Periodic waves Energy transfer via waves on a string
Fig.2. The tension force FT has a component in the direction of the transverse velocity vtr, so at this instant the force is doing work on the end of the string that has a positive value. 2008 by W.H. Freeman and Company Periodic waves Energy transfer via waves on a string
The tension forceFt on the left end of the segment is directed tangent to the string, as shown. To calculate the power transferred by this force, we use the formula , where FT is the tension andvtr , the transverse velocity, is the velocity of the end of the segment. To obtain an expression for the power, we first express the vectors in component form. That is, and Taking the scalar product gives We obtain vy by differentiating Eq.8 From the figure, we see that , where we have used the small angle approximation sintan. Because tan is the slope of line tangent to the string, we have Thus Applying Eq.10 to a harmonic wave (by taking derivatives of Eq.9) gives: (10) Periodic waves Energy transfer via waves on a string
Using and v=/k Eq.6, we substitute for FT and the leading k to obtain: Where v is the wave speed. The average power at any location x is then: Because the average value of cos2(kx-t) is 1/2. This average is taken over an entire period T of the motion with x held constant. The energy travels along a taut string at an average speed equal to the wave speed v, so the average energy (E)av flowing past point P1 during time t (Fig.3) is: The energy is distributed over a length x=vt, so the average energy in length x is: Note that like the average power, the average energy per unit length is proportional to the square of the amplitude of the wave. (11) (12) (13) Periodic waves Energy transfer via waves on a string
Fig.3. The wave has reached point P at time t1. During time t, the wave advanced past point P at a distance vt. 2008 by W.H. Freeman and Company Harmonic Sound Waves Harmonic sound waves can be generated by a tuning fork or loudspeaker that is vibrating with simple harmonic motion. The vibrating source causes the air molecules next to it to oscillate with simple harmonic motion about their equilibrium positions. These molecules collide with neighboring molecules, causing them to oscillate, which in turn collide with their neighboring molecules, causing them to oscillate, and so forth, thereby propagating the sound wave. Eq.5 describes a harmonic sound wave if the wave function y(x,t) is replaced by s(x,t), which represents the displacements of the molecules from their equilibrium positions. Thus, These displacements are along the direction of propagation of the wave, and lead to variations in the density and pressure of the air. Fig. 4 shows the displacement of air molecules and the density variations caused by a sound wave at some fixed time. The pressure is maximum where the density is maximum. We see from this figure that the density wave, and thus the pressure wave, is 90o out of phase with displacement wave. (in the arguments of sine or cosine functions, we will always express phase angles in radians. However, in verbal descriptions, we usually say that: two waves are 90o out of phase, rather than two waves are out of phase by /2 rad. (14) Harmonic Sound Waves Where the displacement s is zero, the density, and thus the pressure, is either maximum or minimum, and where the displacement is a maximum or a minimum, the density, and thus the pressure, is equal to its equilibrium value. A displacement wave given by Eq.14 thus implies a pressure wave given by: Where p stands for the pressure minus the local equilibrium pressure, and p0, the maximum value of p, is called the pressure amplitude. It can be shown that the pressure amplitude p0 is related to the displacement amplitude s0 by: Where v is the speed of propagation and is the equilibrium density of the gas. Thus, as a harmonic sound wave travels through air, the displacement of air molecules, the pressure, and the density all vary sinusoidally with the frequency of the vibrating source. (15) (16) Harmonic Sound Waves Figure 4: a) displacement from equilibrium of air molecules in a harmonic sound wave versus position at some instant. Points x1 and x2 are points of zero displacement. b) some representative molecules equally spaced at their equilibrium position cycle earlier. The arrows indicate the directions of their velocities at that instant. c) molecules near points x1, x2 and x3 after the sound wave arrives. d) density of the air at this instant. e) Pressure change, which is proportional to the density change, versus position. The pressure change and displacement are 90 degrees out of phase. 2008 by W.H. Freeman and Company Energy of sound waves The average energy of a harmonic sound wave in a volume element V is given by Eq.13 with A replaced by s0 and x, replaced by V, where is the equilibrium density of the medium. The energy per unit volume is the average energy density av: (17) (18) Electromagnetic Waves
Waves of energy emitted from any accelerating charges Any object that is above absolute zero emits electromagnetic waves The entire range of possibilities is called the Electromagnetic Spectrum Waves in Three-Dimensions (3D)
Fig.5 shows 2D circular waves on the surface of water in a ripple tank. These waves are generated by drops of water striking the surface. The wave crests from concentric circles called wavefronts. For a point source of sound, the waves move out in 3D, and the wavefronts are concentric spherical surfaces. Fig.5. Circular wavefronts diverging from a point source in a ripple tank. 2008 by W.H. Freeman and Company 2008 by W.H. Freeman and Company Waves in Three-Dimensions
The motion of any set of wavefronts can be indicated by rays, which are directed lines perpendicular to the wavefronts (Fig.6). For circular or spherical waves, the rays are radial lines. Fig.6. The motion of wavefronts can be represented by rays drawn perpendicular to the wavefronts. For a point source, the rays are radial lines diverging from the source. 2008 by W.H. Freeman and Company Waves in Three-Dimensions
In a homogeneous medium, such as air at constant density, the wavefronts travels in straight lines in the direction of the rays, much like a beam of particles. At a great distance from a point source, a sufficiently small section of the wavefront can be approximated by a flat source (a plane), and the rays are approximately parallel lines; such a wave is called a plane wave (Fig.7). The two-dimensional analog of a plane wave is a line wave, which is a small part of a circular wavefront at a great distance from the source. Line waves can also be produced in a ripple tank by a line source, as in Fig.8. Waves in Three-Dimensions
Fig.7. Plane waves. At great distances from a point source, the wavefronts are approximately parallel lines perpendicular to the wavefronts. 2008 by W.H. Freeman and Company Waves in Three-Dimensions
Fig.8. A two-dimensional analog of a plane wave can be generated in a ripple tank by a flat board that oscillates up and down in the water to produce the wavefronts, which are straight lines. 2008 by W.H. Freeman and Company Wave Intensity If a point source emits waves uniformly in all directions, then the energy at a distance r from the source is distributed uniformly on a spherical surface of radius r and area A=4r2. If Pav is the average power emitted by the source is Pav/(4r2). The average power per unit area that is incident perpendicular to the direction of propagation is called the intensity. The SI units of intensity are watts per square meter (W/m2). At a distance r from a point source, the intensity is: The intensity of a three-dimensional wave varies inversely with the square of the distance from a point source. (19) Intensity Defined (20) Intensity due to a point source Wave Intensity There is a simple relation between the intensity of a wave and the energy density in the medium through which it propagates. Fig.9 shows a spherical wave that has just reached the radius r1. The volume inside the radius r1 contains energy because the particles in that region are oscillating. The region outside r1 contains no energy because the wave has not yet reached it. After a short time t, the wave moves out a short distance r=v t past r1. The average energy in the spherical shell of surface area A, thickness v t, and volume V=A r=Av t is: The rate of transfer of energy is the power passing into the shell. The average incident power is And the intensity of the wave is: (21) Wave Intensity Figure 9 2008 by W.H. Freeman and Company Wave Intensity Thus, the intensity equals the product of the wave speed v and the average energy density av. Substituting av=1/22s02 from Eq.18 for the energy density in a harmonic sound wave, we obtain: where we have used s0=p0/(v) from Eq.16 This result-that the intensity of a sound wave is proportional to the square of the amplitude-is a general property of harmonic waves. (22) Wave Intensity The human ear car accommodate a large range of sound-wave intensities, from about 10-12W/m2 (which is usually taken to be the threshold of hearing) to about 1 W/m2 (an intensity great enough to estimate pain in most people). The pressure amplitudes that correspond to these extreme intensities are about 3x10-5 Pa for the hearing threshold and 30 Pa for the pain threshold. (Recall that a Pascal is a Newton per square meter). These very small pressure variations add to or subtract from the normal atmosphere pressure of about kPa. Wave Intensity Fig.10. Sound waves from a telephone handset spreading out in the air. The waves have been made visible by sweeping out the space in front of the handset with a light source whose brightness is controlled by a microphone. 2008 by W.H. Freeman and Company 2008 by W.H. Freeman and Company