Editing Standing wave (section)

=== 2D standing wave with a rectangular boundary ===

Next, consider transverse waves that can move along a two dimensional surface within a rectangular boundary of length ''L<sub>x</sub>'' in the ''x''-direction and length ''L<sub>y</sub>'' in the ''y''-direction. Examples of this type of wave are water waves in a pool or waves on a rectangular sheet that has been pulled taut. The waves displace the surface in the ''z''-direction, with {{nowrap|''z'' {{=}} 0}} defined as the height of the surface when it is still.

In two dimensions and Cartesian coordinates, the [[wave equation]] is
:<math>\frac{\partial^2 z}{\partial t^2}  \;=\; c^2 \left(\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2}\right), </math>

where
*''z''(''x'',''y'',''t'') is the displacement of the surface,
*''c'' is the speed of the wave.

To solve this differential equation, let's first solve for its [[Fourier transform]], with
:<math> Z(x,y,\omega) = \int_{-\infty}^{\infty}z(x,y,t) e^{-i\omega t}dt.</math>

Taking the Fourier transform of the wave equation,
:<math> \frac{\partial^2 Z}{\partial x^2} + \frac{\partial^2 Z}{\partial y^2} = -\frac{\omega^2}{c^2}Z(x,y,\omega). </math>

This is an [[eigenvalues and eigenvectors#Eigenvalues and eigenfunctions of differential operators|eigenvalue]] problem where the frequencies correspond to eigenvalues that then correspond to frequency-specific modes or eigenfunctions. Specifically, this is a form of the [[Helmholtz equation]] and it can be solved using [[separation of variables]].<ref>{{cite web| last=Weisstein| first= Eric W.| title = Helmholtz Differential Equation--Cartesian Coordinates| series = MathWorld--A Wolfram Web Resource| url = https://mathworld.wolfram.com/HelmholtzDifferentialEquationCartesianCoordinates.html| access-date=January 2, 2021}}</ref> Assume
:<math> Z = X(x)Y(y).</math>

Dividing the Helmholtz equation by ''Z'',
:<math> \frac{1}{X(x)}\frac{\partial^2 X}{\partial x^2} + \frac{1}{Y(y)}\frac{\partial^2 Y}{\partial y^2} + \frac{\omega^2}{c^2} = 0. </math>

This leads to two coupled ordinary differential equations. The ''x'' term equals a constant with respect to ''x'' that we can define as
:<math> \frac{1}{X(x)}\frac{\partial^2 X}{\partial x^2} = (ik_x)^2.</math>

Solving for ''X''(''x''),
:<math> X(x) = A_{k_x} e^{i k_x x} + B_{k_x}e^{-i k_x x}.</math>

This ''x''-dependence is sinusoidal–recalling [[Euler's formula]]–with constants ''A''<sub>''k''<sub>''x''</sub></sub> and ''B''<sub>''k''<sub>''x''</sub></sub> determined by the boundary conditions. Likewise, the ''y'' term equals a constant with respect to ''y'' that we can define as
:<math> \frac{1}{Y(y)}\frac{\partial^2 Y}{\partial y^2} = (ik_y)^2 = k_x^2-\frac{\omega^2}{c^2},</math>

and the [[dispersion relation]] for this wave is therefore
:<math> \omega = c \sqrt{k_x^2 + k_y^2}.</math>

Solving the differential equation for the ''y'' term,
:<math> Y(y) = C_{k_y} e^{i k_y y} + D_{k_y}e^{-i k_y y}.</math>

Multiplying these functions together and applying the inverse Fourier transform, ''z''(''x'',''y'',''t'') is a superposition of modes where each mode is the product of sinusoidal functions for ''x'', ''y'', and ''t'',
:<math> z(x,y,t) \sim e^{\pm i k_x x}e^{\pm i k_y y}e^{\pm i \omega t}.</math>

The constants that determine the exact sinusoidal functions depend on the boundary conditions and initial conditions. To see how the boundary conditions apply, consider an example like the sheet that has been pulled taut where ''z''(''x'',''y'',''t'') must be zero all around the rectangular boundary. For the ''x'' dependence, ''z''(''x'',''y'',''t'') must vary in a way that it can be zero at both {{nowrap|''x'' {{=}} 0}} and {{nowrap|''x'' {{=}} ''L''<sub>''x''</sub>}} for all values of ''y'' and ''t''. As in the one dimensional example of the string fixed at both ends, the sinusoidal function that satisfies this boundary condition is
:<math>\sin{k_x x},</math>

with ''k''<sub>''x''</sub> restricted to
:<math>k_x = \frac{n \pi}{L_x}, \quad n = 1, 2, 3, \dots</math>

Likewise, the ''y'' dependence of ''z''(''x'',''y'',''t'') must be zero at both {{nowrap|''y'' {{=}} 0}} and {{nowrap|''y'' {{=}} ''L''<sub>''y''</sub>}}, which is satisfied by
:<math>\sin{k_y y}, \quad k_y = \frac{m \pi}{L_y}, \quad m = 1, 2, 3, \dots</math>

Restricting the wave numbers to these values also restricts the frequencies that resonate to
:<math>\omega = c \pi \sqrt{\left(\frac{n}{L_x}\right)^2 + \left(\frac{m}{L_y}\right)^2}.</math>

If the initial conditions for ''z''(''x'',''y'',0) and its time derivative ''ż''(''x'',''y'',0) are chosen so the ''t''-dependence is a cosine function, then standing waves for this system take the form
:<math> z(x,y,t) = z_{\text{max}}\sin \left(\frac{n\pi x}{L_x}\right) \sin \left(\frac{m\pi y}{L_y}\right) \cos \left(\omega t\right). </math>
:<math> n = 1, 2, 3, \dots \quad m = 1, 2, 3, \dots</math>

So, standing waves inside this fixed rectangular boundary oscillate in time at certain resonant frequencies parameterized by the integers ''n'' and ''m''. As they oscillate in time, they do not travel and their spatial variation is sinusoidal in both the ''x''- and ''y''-directions such that they satisfy the boundary conditions. The fundamental mode, {{nowrap|''n'' {{=}} 1}} and {{nowrap|''m'' {{=}} 1}}, has a single antinode in the middle of the rectangle. Varying ''n'' and ''m'' gives complicated but predictable two-dimensional patterns of nodes and antinodes inside the rectangle.<ref>{{cite AV media | author-last = Gallis| author-first = Michael R.| title = 2D Standing Wave Patterns (rectangular fixed boundaries)| series = Animations for Physics and Astronomy| publisher = Pennsylvania State University| date = February 15, 2008| url = http://phys23p.sl.psu.edu/phys_anim/waves/standingwaves_2D_square.mp4| access-date = December 28, 2020| id = Also available as YouTube Video ID: NMlys8A0_4s}}</ref>

From the dispersion relation, in certain situations different modes–meaning different combinations of ''n'' and ''m''–may resonate at the same frequency even though they have different shapes for their ''x''- and ''y''-dependence. For example, if the boundary is square, {{nowrap|''L''<sub>''x''</sub> {{=}} ''L''<sub>''y''</sub>}}, the modes {{nowrap|''n'' {{=}} 1}} and {{nowrap|''m'' {{=}} 7}}, {{nowrap|''n'' {{=}} 7}} and {{nowrap|''m'' {{=}} 1}}, and {{nowrap|''n'' {{=}} 5}} and {{nowrap|''m'' {{=}} 5}} all resonate at
:<math>\omega = \frac{c \pi}{L_x} \sqrt{50}.</math>

Recalling that ''ω'' determines the eigenvalue in the Helmholtz equation above, the number of modes corresponding to each frequency relates to the frequency's [[eigenvalues and eigenvectors#Eigenspaces, geometric multiplicity, and the eigenbasis|multiplicity]] as an eigenvalue.