Part 2: Making the connection between mechanical waves on a string and Electromagnetic Waves. (6 pts) For any mechanical wave the wavefunction contains all the information about the propagation speed and direction of the wave. The wave speed is set by the medium, which was represented by the tension in the string. Chapters 14, 15, and 16 in your text develops of the wavefunction starting with the simple harmonic motion (Ch 14) of a mass oscillating on a spring about its equilibrium position. The mathematics also applies to a pendulum swinging back and forth and many other phenomena. The equation only had a time variable, t, and was described equally well by either a sine or a cosine function. For the mass on the spring, we generally applied a force to displace the mass by stretching the spring then let it go. Because we started at the maximum displacement at time t = 0, we chose the cosine function to describe the motion: x(t) = Acos(mt) = Acos (gt) = Acos(2uft) They also discussed the velocity and acceleration of the object that was oscillating and showed that they were also sinusoidal functions and showed the similarities both graphically and mathematically, taking successive slopes of the graph, or time derivatives of the function, led to 15,0) = wAsin(cut); and ax(t) = cu2Acos(wt) Chapter 15 introduced the wave model, which included transverse and longitudinal waves, with the addition of the spatial variable (x, y, or 2) added to the inside of the sinusoidal function. Traveling waves all have 2 distinct motions l) the back/forth, up/down, or side to side oscillation of the medium, which is described by the oscillatory sine or cosine functions and their derivatives, and 2) the propagation of the wave carrying energy through the medium. In all these cases the medium determines the propagation speed, v, with a similar form: restoring force returning system to equilibrium _) l7 _ T. l) _ 'ykgT _ yRT inertia resisting return to equilibrium string 11' sound m M For waves on a string, T is the tension (in newtons, N) and ,u is the mass per unit length of the string (in kg/m), both physical properties of the string. For sound waves in a gas, T is the absolute temperature of the gas (in kelvin), R is the ideal gas constant, M is the molar mass, and the parameter 3) is a constant that depends on whether the gas is monoatomic, diatomic, or other. Again, these all describe the physical properties of the medium. Sound also travels though liquids and solids, and when it does there is a similar ratio with parameters that describe the bulk properties (density, elasticity, etc.) of the medium. In all cases, the waves could be described by the same wave inction: \"medium vxu) = wASin(wt); and axU) = cu'Acos(wt) Chapter 15 introduced the wave model, which included transverse and longitudinal waves, with the addition of the spatial variable (x, y, or 2) added to the inside of the sinusoidal function. Traveling waves all have 2 distinct motions l) the back/forth, up/down, or side to side oscillation of the medium, which is described by the oscillatory sine or cosine functions and their derivatives, and 2) the propagation of the wave carrying energy through the medium. In all these cases the medium determines the propagation speed, v, with a similar form: restoring force returning system to equilibrium T ykg'l' yRT - - - - ' - - _) vstrin = _; \"sound = = inertia reSisting return to equilibrium 9 pi m M For waves on a string, T is the tension (in newtons, N) and ,u is the mass per unit length of the string (in kg/m), both physical properties of the string. For sound waves in a gas, T is the absolute temperature of the gas (in kelvin), R is the ideal gas constant, M is the molar mass, and the parameter y is a constant that depends on whether the gas is monoatomic, diatomic, or other. Again, these all describe the physical properties of the medium. Sound also travels though liquids and solids, and when it does there is a similar ratio with parameters that describe the bulk properties (density, elasticity, etc.) of the medium. In all cases, the waves could be described by the same wave function: vmedium y(x, t) = Acos(kx wt) or y(x. t) = Asin(kx cut). The wave speed and the propagation direction are \"hidden\" in the coefcients in front of the spatial (x) and temporal (1:) variables. Of course, all waves are not limited to propagating along the x-axis and having their amplitudes along the y-axis, but we can extract that information from the wavefunction itself, no matter what order the space and time variables are written or whether the function is a sine or cosine, the spatial direction is always the spatial variable inside the sinusoidal function (x, y, or z) and its coefcient is always k = 21,\" and the coefcient in front of the time 211? variable (t) is always to = 2n f = T Hence, from the wavefunction we can extract the wave speed v, the propagation axis through space, and we can tell if the wave is propagating in the positive, +, or negative, , x, y, or 2, directions by the relative signs in front of each term. If both signs are the same, the wave propagates in the negative direction, if the two terms have different signs, it does not matter which, the wave propagates in the positive direction. This is true for any mechanical wave in any medium. Given the following wave function, determine the amplitude A, wavenumber k, wavelength ,1, frequency f: angular frequency a), period T, and propagation direction of the wave (as d: x, y, or z). (6 pts) y(z,t) = (3.75 m) cos[(42.5 rad/m)z + (98.6 rad/s)t]