Αποτελέσματα Αναζήτησης
K1. Vocabulary: amplitude, wavelength, wave number, phase, phase constant, wave function, wave speed, wave equation, harmonic function, sinusoidal wave, traveling wave, boundary conditions, ̄eld.
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. At what speed will this wave travel? The speed of a wave is 65 m/sec.
2. Sinusoidal waves We have seen that the wave equation is solved by the d’Alembert solution y(x;t) = f(x ct) + g(x+ ct). A particularly interesting option for f(u) and g(v) are sines and cosines. For example, we can choose f(x ct) = Ccos(k(x ct) + ’): (2.1) The sinusoidal wave is charaterised by Wavenumber = k, Wavelength = 2ˇ=k,
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
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.
EXAMPLE PROBLEMS. Find the height, the length, the period, the celerity (=wave speed) of this wave, and in which direction it goes, where η and x are in meters, and t in seconds. 𝜂𝜂𝑥𝑥,𝑡𝑡= cos 𝑥𝑥−𝑡𝑡. Do the same if: 𝜂𝜂𝑥𝑥,𝑡𝑡= cos 𝑥𝑥+𝑡𝑡. What about if:
First, estimate the number of wave crests that pass a given point per second. This is the frequency of the wave. Then, estimate the distance between two successive crests, which is the wavelength. The product of the frequency and the wavelength is the speed of the wave. 7.
