Wave Equation Basics

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### Introduction to Wave Equations Wave equations describe the propagation of various types of waves, such as sound waves, light waves, and water waves. They are fundamental in physics and engineering. ### General Form of the Wave Equation The one-dimensional wave equation for a displacement function $u(x, t)$ is given by: $$\frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial x^2}$$ Where: - $u(x, t)$ is the displacement of the wave at position $x$ and time $t$. - $v$ is the wave speed. ### Properties of Waves - **Amplitude (A):** Maximum displacement from equilibrium. - **Wavelength ($\lambda$):** Spatial period of the wave, distance between two consecutive crests or troughs. - **Frequency (f):** Number of complete oscillations per unit time. - **Period (T):** Time taken for one complete oscillation ($T = 1/f$). - **Wave Speed (v):** The speed at which the wave propagates. $v = f\lambda = \lambda/T$. - **Angular Frequency ($\omega$):** $\omega = 2\pi f$. - **Wave Number (k):** $k = 2\pi/\lambda$. The wave speed can also be expressed as $v = \omega/k$. ### Solutions to the Wave Equation A general solution to the one-dimensional wave equation is of the form: $$u(x, t) = f(x - vt) + g(x + vt)$$ Where: - $f(x - vt)$ represents a wave traveling in the positive x-direction. - $g(x + vt)$ represents a wave traveling in the negative x-direction. For a sinusoidal wave, common forms include: - $u(x, t) = A \sin(kx - \omega t + \phi)$ - $u(x, t) = A \cos(kx - \omega t + \phi)$ Where $\phi$ is the phase constant. ### Transverse Waves In transverse waves, the oscillations are perpendicular to the direction of wave propagation. - **Examples:** Light waves, waves on a string. - For a string under tension $T$ and linear mass density $\mu$, the wave speed is: $$v = \sqrt{\frac{T}{\mu}}$$ ### Longitudinal Waves In longitudinal waves, the oscillations are parallel to the direction of wave propagation. - **Examples:** Sound waves. - For sound waves in a fluid with bulk modulus $B$ and density $\rho$, the wave speed is: $$v = \sqrt{\frac{B}{\rho}}$$ - For sound waves in a solid rod with Young's modulus $Y$ and density $\rho$, the wave speed is: $$v = \sqrt{\frac{Y}{\rho}}$$ ### Superposition Principle When two or more waves overlap, the resultant displacement at any point is the algebraic sum of the displacements due to individual waves. $$u_{total}(x, t) = u_1(x, t) + u_2(x, t) + ...$$ This principle leads to phenomena like interference and diffraction. ### Standing Waves Standing waves are formed when two waves of the same frequency, amplitude, and wavelength traveling in opposite directions interfere. - They have fixed nodes (points of zero displacement) and antinodes (points of maximum displacement). - For a string fixed at both ends of length L, the allowed wavelengths are: $$\lambda_n = \frac{2L}{n} \quad \text{for } n = 1, 2, 3, ...$$ - The corresponding frequencies are: $$f_n = \frac{v}{\lambda_n} = \frac{nv}{2L}$$ Where $f_1$ is the fundamental frequency (first harmonic), $f_2 = 2f_1$ is the second harmonic, and so on. ### Doppler Effect The apparent change in frequency of a wave due to the relative motion between the source and the observer. - For sound waves: $$f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right)$$ Where: - $f'$ is the observed frequency. - $f$ is the source frequency. - $v$ is the speed of sound in the medium. - $v_o$ is the speed of the observer. - $v_s$ is the speed of the source. **Sign Convention:** - Use '+' for $v_o$ if observer moves towards source, '-' if away. - Use '-' for $v_s$ if source moves towards observer, '+' if away.