MathWorks is the only company in the world whose logo satisfies a partial differential equation. Why is the region for this equation shaped like a capital letter L?

### Contents

- Wave Equation
- Initial and Boundary Values
- Eigenvalues and Eigenfunctions
- One Dimension
- A Square
- A Circular Disc
- A Circular Sector
- The L-shaped Membrane
- Microwave Waveguide

#### Wave Equation

The wave equation describes how a disturbance travels through matter. If the units are chosen so that the wave propagation speed is equal to one, the amplitude of a wave satisfies$$ {{\partial^2 u} \over {\partial t^2}} = \triangle u $$The $\triangle$ denotes Laplace's operator$$ \triangle = {\partial^2 \over {\partial x^2}}+ {\partial^2 \over {\partial y^2}} $$

#### Initial and Boundary Values

Geometry plays a crucial role here. Initial values of the amplitude and velocity of the wave are prescribed on a certain region. Values of the amplitude or its normal derivative are also prescribed on the boundary of the region. If the wave vanishes outside the region, these boundary values are zero.

#### Eigenvalues and Eigenfunctions

Separating out periodic time behavior leads to solutions of the form$$ u(x,y,t) = \cos{(\sqrt{\lambda}\,t)} v(x,y) $$The functions $v(x,y)$ also depend upon $\lambda$ and the region. They satisfy the differential equation$$ \triangle v + \lambda v = 0 $$and are zero on the boundary of the region. The quantities $\lambda$ that lead to nonzero solutions are the *eigenvalues*, and the corresponding functions $v(x,y)$ are the *eigenfunctions* or *modes*. They are determined by the physical properties of the medium and the geometry of the region. The square roots of the eigenvalues are resonant frequencies. A periodic external driving force at one of these frequencies generates an unboundedly strong response in the medium.Any solution of the wave equation can be expressed as a linear combination of these eigenfunctions. The coefficients in the linear combination are obtained from the initial conditions.

#### One Dimension

In one dimension, the eigenvalues and eigenfunctions are easily determined. The simplest example is a violin string, held fixed at the ends of an interval. The eigenfunctions are trig functions.$$ v_k(x) = \sin{(k x)} $$If the length of the interval is $\pi$, the eigenvalues are determined by the boundary condition, $v_k(k \pi) = 0$. Hence, $k$ must be an integer and$$ \lambda_k = k^2 $$If the initial condition is expanded in a Fourier sine series,$$ u(x,0) = \sum_k a_k \sin{(k x)} $$(And the initial velocity is zero), then the solution to the wave equation is$$ u(x,t) = \sum_k a_k \cos{(\sqrt{\lambda_k}\,t)} v_k(x) $$Here are graphs of the first nine eigenfunctions in one dimension. The corresponding eigenvalues are the squares of integers.

` eigenvals = (1:9).^2 plot_modes('1d')`

eigenvals = 1 4 9 16 25 36 49 64 81

#### A Square

The simplest region in two dimensions is a square. The eigenfunctions are again trig functions.$$ v_{k,j}(x,y) = \sin{(k x)}\,\sin{(j y)} $$If the sides have length $\pi$, the boundary conditions imply that $k$ and $j$ must be integers. Here are the first nine eigenvalues and eigenfunctions.

` [k,j] = meshgrid(1:3); e = k.^2+j.^2; eigenvals = e(:)' plot_modes('square')`

eigenvals = 2 5 10 5 8 13 10 13 18

#### A Circular Disc

If the region is a circular disc, we switch to polar coordinates, $r$ and $\theta$. Trig functions are replaced by Bessel functions. The eigenfunctions become$$ v_{k,j}(r,\theta) = B_j(\mu_k r) \,\sin{(j \theta)} $$where $B_j$ is the $j$ -th order Bessel function and $\mu_k = \sqrt{\lambda_k}$. To find the eigenvalues we need to have the eigenfunctions vanish on the boundary of the disc. If the radius is one, we require$$ B_j(\mu_k) = 0 $$In other words, we need to compute zeros of Bessel functions. Here are the first nine eigenvalues and eigenfunctions of the circular disc. The violin string has become a tambourine.

` eigenvals = [bjzeros(0,3) bjzeros(1,3) bjzeros(2,3)].^2 plot_modes('disc')`

eigenvals = Columns 1 through 7 5.7832 30.4713 74.8870 14.6820 49.2185 103.4995 26.3746 Columns 8 through 9 70.8500 135.0207

#### A Circular Sector

Replace the full circular disc by a three-quarter circular sector. The angle at the origin is $270^\circ$ or $\frac{3}{2}\pi$ radians. We can make our eigenfunctions adapt to this angle. Take$$ v_{k,j}(r,\theta) = B_{\alpha_j}(\mu_k r) \,\sin{(\alpha_j \theta)} $$with$$ \alpha_j = \frac{2}{3} j $$and fractional order Bessel functions. The eigenfunctions satisfy their differential equation and also satisfy the boundary conditions on both sides of the angle.$$ v_{k,j}(r,\theta) = 0 $$at $\theta = 0$ and at $\theta = \frac{3}{2}\pi$.By finding the zeros of the Bessel functions we can also have the eigenfunctions satisfy the boundary conditions on the outer circular portion of the boundary. Here are the first nine eigenvalues and eigenfunctions of the three-quarter circular sector.

` eigenvals = [bjzeros(2/3,3) bjzeros(4/3,3) bjzeros(6/3,3)].^2 plot_modes('sector')`

eigenvals = Columns 1 through 7 11.3947 42.6442 93.6362 18.2785 56.1131 113.6860 26.3746 Columns 8 through 9 70.8500 135.0207

These eigenfunctions have another important property. Most of them are singular; the derivatives of the fractional order Bessel functions are unbounded at the origin. You can see the black concentration of grid lines in the plots. If you tried to make a tambourine with this sector shape, it would rip at the sharp corner. This singular behavior is needed to model the solution to the wave equation on this region.

#### The L-shaped Membrane

For all the regions we have discussed so far it is possible to express the eigenvalues as zeros of analytic functions. For the interval and the square, the eigenvalues are integers, which are the zeros of $\sin{\pi x}$. For the circular disc and sector, the eigenvalues are zeros of Bessel functions. Once we have the eigenvalues, it is easy to compute the eigenfunctions using sines and Bessel functions.So, an L-shaped region formed from three unit squares is interesting for at least two reasons. It is the simplest geometry for which solutions to the wave equation cannot be expressed analytically; numerical computation is necessary. Furthermore, the 270 degree nonconvex corner causes a singularity in the solution. Mathematically, the gradient of the first eigenfunction is unbounded near the corner. Physically, a membrane stretched over such a region would rip at the corner. This singularity limits the accuracy of finite difference methods with uniform grids.I used the L-shaped region as the primary example in my doctoral thesis fifty years ago. MathWorks has adopted a modified surface plot of the first eigenfunction as the company logo. I am going to devote a series of blog posts to the L.Here are the first nine eigenvalues and eigenfunctions, computed by a function from Numerical Computing with MATLAB, which I will discuss in a future posting. Compare these eigenfunctions with the ones from the circular sector, which has the same reentrant corner and resulting singularity.

for k = 1:9 [~,eigenvals(k)] = membranetx(k); end eigenvals plot_modes('L')

eigenvals = Columns 1 through 7 9.6397 15.1973 19.7392 29.5215 31.9126 41.4745 44.9485 Columns 8 through 9 49.3480 49.3480

#### Microwave Waveguide

Simple model problems involving waves on an L-shaped region include an L-shaped membrane, or L-shaped tambourine, and a beach towel blowing in the wind, constrained by a picnic basket on one fourth of the towel.A more practical example involves ridged microwave waveguides. One such device is a waveguide-to-coax adapter. The active region is the channel with the H-shaped cross section visible at the end of the adapter. The ridges increase the bandwidth of the guide at the expense of higher attenuation and lower power-handling capability. Symmetry of the H about the dotted lines shown in the contour plot of the electric field implies that only one quarter of the domain needs to be considered and that the resulting geometry is our L-shaped region. The boundary conditions are different than our membrane problem, but the differential equation and the solution techniques are the same.The photo is courtesy of Advanced Technical Materials, Inc. See their website, <http://atmmicrowave.com/>, for lots of devices like this.

waveguide

Published with MATLAB® R2014b

## FAQs

### What is the shape of the Matlab logo? ›

The MathWorks logo is a lighted, reflective, surface plot of a variant of an eigenfunction of the L-shaped membrane. While the logo itself is a familiar icon, its mathematical background is less well known.

**What is the meaning of the Matlab logo? ›**

The MATLAB logo contains no wordmark or framing, it is just a pure form, which is colored orange with the addition of a metallic gradient. The orange color is commonly known as **a symbol of creativity and energy**. It also shows the passion of the company in what it does.

**What is the Lagrange symbol in MATLAB? ›**

The structure is called lambda because the conventional symbol for Lagrange multipliers is the **Greek letter lambda (λ)**.

**How do i get the shape of an image in MATLAB? ›**

**Direct link to this comment**

- % Open, Read, & Display Image.
- baseFileName = 'F:\PhD\MATLAB CODING\BlobsDemo\shape. ...
- rgbImg = imread(baseFileName);
- [rows, columns, numberOfColorBands] = size(rgbImg);
- subplot(2, 2, 1);
- imshow(rgbImg);
- set(gcf, 'Position', get(0,'Screensize')); % Enlarge figure to full screen.

**What is the symbol for or in MATLAB? ›**

The symbols | and || perform different operations in MATLAB^{®}. The element-wise OR operator described here is | . The short-circuit OR operator is || .

**What is symbol function in MATLAB? ›**

Symbolic functions **accept array inputs**. Calculate f for multiple values of x and y . You can differentiate symbolic functions, integrate or simplify them, substitute their arguments with values, and perform other mathematical operations. For example, find the derivative of f(x,y) with respect to x .

**How is an image represented in MATLAB? ›**

In the MATLAB workspace, most images are represented as **two-dimensional arrays (matrices)**, in which each element of the matrix corresponds to a single pixel in the displayed image. For example, an image composed of 200 rows and 300 columns of different colored dots stored as a 200-by-300 matrix.

**What is Lambda symbol Lagrange? ›**

The Lagrange multiplier, λ, **measures the increase in the objective function (f(x, y) that is obtained through a marginal relaxation in the constraint (an increase in k)**. For this reason, the Lagrange multiplier is often termed a shadow price.

**What does the Lagrangian function represent? ›**

Lagrangian function, also called Lagrangian, quantity that **characterizes the state of a physical system**. In mechanics, the Lagrangian function is just the kinetic energy (energy of motion) minus the potential energy (energy of position).

**What is Lagrange in math? ›**

The Lagrange method of multipliers is named after Joseph-Louis Lagrange, the Italian mathematician. The primary idea behind this is to **transform a constrained problem into a form so that the derivative test of an unconstrained problem can even be applied**.

### How do I shape an image into a shape? ›

In your file, select the picture that you want to crop. On the Picture Format tab, click the arrow next to Crop. (If you don't see the Picture Format tab, make sure that you've selected a picture (not a shape).) Point to Crop to Shape and then click the shape you want to crop to.

**What is shape factor MATLAB? ›**

Shape factor — **RMS divided by the mean of the absolute value**. Shape factor is dependent on the signal shape while being independent of the signal dimensions. The higher-order statistics provide insight to system behavior through the fourth moment (kurtosis) and third moment (skewness) of the vibration signal.

**What is shape recognition? ›**

Preattentive shape recognition **allows quick analysis of shapes and provides useful dimensions for comprehensible visualization**.

**Does MATLAB use i or J? ›**

In MATLAB^{®}, **i and j represent the basic imaginary unit**. You can use them to create complex numbers such as 2i+5 . You can also determine the real and imaginary parts of complex numbers and compute other common values such as phase and angle.

**What does == mean in MATLAB? ›**

A == B **returns a logical array or a table of logical values with elements set to logical 1 ( true ) where inputs A and B are equal**; otherwise, the element is logical 0 ( false ).

**What is the infinity symbol in MATLAB? ›**

Infinity Symbol, **inf**

Use the MATLAB^{®} symbol inf to represent infinity in C charts. Calculations like n/0 , where n is any nonzero real value, result in inf .

**How to solve symbolic function in MATLAB? ›**

**S = solve( eqn , var )** solves the equation eqn for the variable var . If you do not specify var , the symvar function determines the variable to solve for. For example, solve(x + 1 == 2, x) solves the equation x + 1 = 2 for x.

**What is symbol function? ›**

A symbol function is **used to set a symbol value depending on the parameters provided for the function**.

**Can you put symbols in MATLAB? ›**

**By default, MATLAB ^{®} supports a subset of TeX markup**. To use additional special characters, such as integral and summation symbols, you can use LaTeX markup instead. This example shows how to insert Greek letters, superscripts, and annotations into chart text and explains other available TeX options.

**How is an image represented? ›**

**An image is made of many pixels, each of them with a particular color**. Modern computers can show up to 16.7 millions of different colors. It is impossible to store each one of these colors personnally, they are instead represented as a combination of three primary colors: red, green and blue (RGB color model).

### Is White 0 or 1? ›

These pixel values represent the intensity of each pixel. 0 represents black and **255 represents white**.

**How to check details of image in MATLAB? ›**

Get the image metadata. **info = imfinfo('bag.** **png');** Open an Image Information tool associated with the figure that also contains the image metadata.

**What is the structure of MATLAB? ›**

Structures and cell arrays are two kinds of MATLAB arrays that can hold generic, unstructured heterogeneous data. **A structure array is a data type that groups related data using data containers called fields**.

**What is the degree symbol figure in MATLAB? ›**

**char(176)** is the degree symbol in MATLAB..

**What is square in MATLAB? ›**

square is **similar to the sine function but creates a square wave with values of –1 and 1**. x = square( t , duty ) generates a square wave with specified duty cycle duty . The duty cycle is the percent of the signal period in which the square wave is positive.

**Is MATLAB based on C or C++? ›**

John Little and programmer Steve Bangert re-programmed MATLAB in **C**, created the MATLAB programming language, and developed features for toolboxes.

**What is MATLAB basic concept? ›**

Matlab Basics

**MATLAB is designed to work with matrices, where a matrix is defined to be a rectangular array of numbers**. All variables used are considered to be matrices. Scalars and vectors can be used since they can be considered as matrices with dimension 1×1 (scalars) and 1xn or nx1 (vectors).

**What is MATLAB explained? ›**

MATLAB^{®} is **a programming platform designed specifically for engineers and scientists to analyze and design systems and products that transform our world**. The heart of MATLAB is the MATLAB language, a matrix-based language allowing the most natural expression of computational mathematics.

**What is J or i in MATLAB? ›**

**You can use j to enter complex numbers.** **You also can use the character i as the imaginary unit**. To create a complex number without using i and j , use the complex function. z = a + b j returns a complex numerical constant, z .

**Can you use symbols in MATLAB? ›**

By default, MATLAB^{®} supports a subset of TeX markup. **To use additional special characters, such as integral and summation symbols, you can use LaTeX markup instead**. This example shows how to insert Greek letters, superscripts, and annotations into chart text and explains other available TeX options.

### Is MATLAB in degrees or radians? ›

Degrees and Radians

**Many Mapping Toolbox functions, such as distance and azimuth , use degrees by default and allow you to choose radians**. Some functions, such as unwrapMultipart and meridianarc , use radians by default or require you to work in radians.

**What is the degree symbol icon? ›**

The degree symbol or degree sign, °, is a glyph or symbol that is used, among other things, to represent degrees of arc (e.g. in geographic coordinate systems), hours (in the medical field), degrees of temperature or alcohol proof. The symbol consists of **a small superscript circle**.

**What is the degree symbol code? ›**

Insert the degree symbol by using a keyboard shortcut

On your keyboard, press **Alt + 0176**.

**How do you write 2 squares in Matlab? ›**

For example, you might write x. ^2 in another way, using x. *x. This would effectively square every element in the vector x.

**How do you draw a circle in Matlab? ›**

**Direct link to this answer**

- function h = circle(x,y,r)
- hold on.
- th = 0:pi/50:2*pi;
- xunit = r * cos(th) + x;
- yunit = r * sin(th) + y;
- h = plot(xunit, yunit);
- hold off.

**What is axis square in Matlab? ›**

axis square **makes the current axes region square (or cubed when three-dimensional)**. This option adjusts the x-axis, y-axis, and z-axis so that they have equal lengths and adjusts the increments between data units accordingly.