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Write an equation for the wave as a function of position and time. A wave is modeled with the function y (x, t) = (0.25 m) cos (0.30 m −1 x − 0.90 s −1 t + π 3). Find the (a) amplitude, (b) wave number, (c) angular frequency, (d) wave speed, (e) initial phase shift, (f) wavelength, and (g) period of the wave.
A sinusoidal wave travels down a taut, horizontal string with a linear mass density of μ = 0.060 kg/m. The maximum vertical speed of the wave is v y max = 0.30 cm/s. The wave is modeled with the wave equation y (x, t) = A sin (6.00 m −1 x − 24.00 s −1 t).
b. The equation for a standing wave is: \[\begin{aligned} D(x,t)=2A\sin(kx)cos(\omega t)\end{aligned}\] We let the fixed end be at \(x=0\). At the fixed end, the displacement is equal to zero. At the free end (\(x=L\)) the displacement is maximized. The first condition is always true. The second condition will be met when:
Write down the solution of the wave equation u tt = u xx with ICs u (x; 0) = f (x) and u t (x; 0) = 0 using D'Alembert's formula. Illustrate the nature of the solution by sketching the ux -pro les y = u (x; t) of the string displacement for t = 0 ; 1=2; 1; 3=2. Solution: D'Alembert's formula is 1 Z x+t
Problems for you to try: Complete the following practice problems. You MUST show ALL the work outlined in the steps in the example problems. A wave with a frequency of 14 Hz has a wavelength of 3 meters.
Both the sinusoidal wave and the complex exponential are solutions to the wave equation, and can be used inter-changeably. Why would we focus on sinusoidal solutions to the wave equation? The most important reason is that one can construct any function using them.
Sinusoidal waves render the mathematical analysis, in terms of differential equations, straightforward and, for linear systems, provide a complete basis set from which any solution can be formed as a superposition – even if the system shows dispersion.
