The **wave equation ** is a typical example of more general class of partial differential equations called hyperbolic equations. They occur in classical physics, geology, acoustics, electromagnetics, and fluid dynamics. Wave equations usually describe wave propagations in different media.
Historically, the problem of a vibrating string such as that of a musical instrument was first studied by the French mathematician, mechanician, physicist, philosopher, and music theorist Jean le Rond
d'Alembert.
Because he was the first who found a solution of one-dimensional wave equation in 1746, the latter is usually referred to as **d'Alembert's equation**.
Many others contributed to study of the wave equation, among first of them we mention Leonhard Euler (who discovered the wave equation in three space dimensions), Daniel Bernoulli ( the Eulerâ€“Bernoulli beam equation), and Joseph-Louis
Lagrange (classical and celestial mechanics).

The wave equation for real-valued function \( u(x_1, x_2, \ldots , x_n , t) \) of *n* spatial variables and a time variable *t* is

*c*is a positive constant (having dimensions of speed) and

**d'Alembertian**and △ is the

**Laplacian**.

Suppose we have a medium whose displacement may be described by a scalar function *u*(**x**,*t*), where \( {\bf x} \in \mathbb{R}^n , \quad t\in\mathbb{R} . \) Suppose that the system is conservative and it has the Lagrangian \( {\cal L} = \mbox{K} - \Pi , \) where the kinetic energy K and potential energy Π of the medium are

**x**) is a mass-density and k(

**x**) is a stiffness, both assumed positive, and \( u_t = \partial u/\partial t . \) The corresponding action becomes

*u*(

**x**,

*t*), obtained by integrating the Lagrangian with respect to time.

The Euler--Lagrange equation is satisfied by a stationary point (which is a function *u*(**x**, *t*)) of this action becomes

*k*are constants, then we get the wave equation

We derive the wave equation in one space dimension that models the
transverse vibrations of an elastic string. If such string is placed
horizontally between end points *x=0* and *x*=ℓ, it can
freely vibrate within a vertical plane. Generally speaking it is not
true; however, if displacements *u(x,t)* are small, we can assume
that spring motion occur only within a plane perpendicular to its
equilibrium horizontal position.

Perhaps the easiest case is observed with the investigation of
mechanical vibrations. Suppose that an elastic string of length ℓ
is tightly stretched between two supports at the same horizontal
level, which we identify with *x*-axis. Then its end points may
be taken as *x=0* and *x*=ℓ. The elastic string may be
thought of as a guitar or violin string, a guy wire, or possibly an
electric power line. The positions of points on the string can be
described by the displacement, which we denote by *u(x,t)*, from
the equilibrium horizontal position. If damping effects, such as air
resistance, are neglected, and if the magnitude of the motion is not
too large, then the displacement function satisfies the partial
differential equation (called one dimensional wave equation)

*x*< ℓ 0 <

*t*< ∞. The constant coefficient

*c*² is given by

*T*is the tension (force) in the string, and ρ is the mass per unit length of the string material (density). To describe the motion of the string completely, we need to impose some auxiliary conditions. Of these, we need to specify the initial displacement and its initial velocity

*d*and

*v*are known functions. If we consider a ideal (and not realistic) case that the string has an infinite length, we arrive at so called the

**initial value problem**:

The one dimensional d'Alembertian operator can be recomposed into the product of the first order differential operators:

*u*

_{tt}-

*c*²

*u*

_{xx}= 0 to two first order equations

*x*±

*ct*, each of the above equations is reduced to a simple ordinary differential equation

*u*

_{tt}-

*c*²

*u*

_{xx}= 0 is the sum

*f*(ξ) and

*g*(ξ) of one variable. This formula represents a superposition of two waves, one traveling to the right and another traveling to the left, each with velocity

*x*. However, in practice, traveling waves are excited by the initial disturbance

*d(x)*is the initial displacement (initial configuration) and

*v(x)*is the initial velocity of the string. Upon substituting the general solution into the initial condition, we get two equations

**d'Alembert's formula**)

Let us consider the wave equation in semi-infinite domain 0
< *x* < ∞ For simplicity, we first assume that the
boundary condition at left end *x* = 0 are of first type
(Dirichlet):

We are now in a position to solve the general initial boundary value problem for the wave equation subject to homogeneous boundary conditions of the first type (Dirichlet's conditions are chosen for simplicity):

*X*(0) = 0 and

*X*(ℓ) = 0. The differential equation together with these boundary conditions constitute the Sturm--Liouville problem:

*X*(0) = 0 and

*X*(ℓ) = 0 indicate that

*c*

_{1}= 0 and \( c_2 \sin \alpha \ell =0 . \) The last equation implies that \( \sin \alpha \ell =0 \) because otherwise

*c*

_{2}= 0 forces to obtain a trivial solution. Since we cannot effort vanishing

*c*

_{2}, we have to choose α to be

*t*= 0, we get from the initial condition

*u(x,0) = d(x)*that

*d(x)*, we can write

*B*

_{n}, we differentiate the general solution with respect to

*t*and the set

*t*= 0:

*v(x)*, the total coefficient of sine must be given by

*c*appearing in the solution of the initial boundary value problem is given by \( c= \sqrt{T/\rho} , \) where ρ is mass per unit length and

*T*is the magnitude of the tension in the spring. When

*T*is large enough, the vibrating string produces a musical sound. This sound, according to the solution

**standing waves**or

**normal modes**.

*u*

_{n}

*(x,t)*that at a fixed value of

*x*this function represents simple harmonic motion with amplitude \( C_n \sin \frac{n\pi x}{\ell} \) and frequency

*f*

_{n}=

*nc/(2ℓ)*. In other words, each point on a standing wave vibrates with a different amplitude but with the same frequency. The mode corresponding

*n*= 1,

**first standing wave**, the

**first normal mode**, or the

**fundamental mode of vibration**, and its frequency is called the

**fundamental frequency**:

*f*

_{n}of the other normal modes, which are integer multiples of the fundamental frequency, are called

**overtones**.

Standing wave patterns are wave patterns produced in a medium when two waves of identical frequencies interfere in such a manner to produce points along the medium that always appear to be standing still. These points that have the appearance of standing still are referred to as nodes. There are several frequencies with which the snakey can be vibrated to produce the patterns. Each frequency is associated with a different standing wave pattern. These frequencies and their associated wave patterns are referred to as harmonics. The two individual waves are drawn in blue and green and the resulting shape of the medium is drawn in black.

Solve the initial boundary value problem:

Suppose we pluck a string by pulling it upward and release it from rest (see Fig.\;\ref{F555.1}). If the point of the pluck is in the third of a string of length $\ell$ (which is usually the case when playing guitar), we can model the vibration of the string by solving the following initial boundary value problem:

```
syms x n t
y1 = (x./50).*sin((n.*pi.*x)./120);
y2 = (120-x./70).*sin((n.*pi.*x)./120);
I1 = int(y1,x,0,50);
I2 = int(y2,x,50,120);
Cn = ((1/60).*I1)+((1/60).*I2);
utemp =
subs(Cn.*sin(n.*pi.*x.*(1/120)).*cos(n.*pi.*t.*(1/40)),n,1:10);
U = sum(utemp);
U = subs(U,t,1);
U = subs(U,x,0:120);
plot(0:120,U)
```

picture:
```
syms x n t
y1 = (x./50).*sin((n.*pi.*x)./120);
y2 = (120-x./70).*sin((n.*pi.*x)./120);
I1 = int(y1,x,0,50);
I2 = int(y2,x,50,120);
Cn = ((1/60).*I1)+((1/60).*I2);
utemp =
subs(Cn.*sin(n.*pi.*x.*(1/120)).*cos(n.*pi.*t.*(1/40)),n,1:10);
U = sum(utemp);
U = subs(U,t,5);
U = subs(U,x,0:120);
plot(0:120,U)
```

picture:
```
syms x n t
y1 = (x./50).*sin((n.*pi.*x)./120);
y2 = (120-x./70).*sin((n.*pi.*x)./120);
I1 = int(y1,x,0,50);
I2 = int(y2,x,50,120);
Cn = ((1/60).*I1)+((1/60).*I2);
utemp =
subs(Cn.*sin(n.*pi.*x.*(1/120)).*cos(n.*pi.*t.*(1/40)),n,1:10);
U = sum(utemp);
U = subs(U,t,15);
U = subs(U,x,0:120);
plot(0:120,U)
```

picture:
```
syms x n t
y1 = (x./50).*sin((n.*pi.*x)./120);
y2 = (120-x./70).*sin((n.*pi.*x)./120);
I1 = int(y1,x,0,50);
I2 = int(y2,x,50,120);
Cn = ((1/60).*I1)+((1/60).*I2);
utemp =
subs(Cn.*sin(n.*pi.*x.*(1/120)).*cos(n.*pi.*t.*(1/40)),n,1:10);
U = sum(utemp);
U = subs(U,t,25);
U = subs(U,x,0:120);
plot(0:120,U)
```

picture:
```
syms x n t
y1 = (x./50).*sin((n.*pi.*x)./120);
y2 = (120-x./70).*sin((n.*pi.*x)./120);
I1 = int(y1,x,0,50);
I2 = int(y2,x,50,120);
Cn = ((1/60).*I1)+((1/60).*I2);
utemp =
subs(Cn.*sin(n.*pi.*x.*(1/120)).*cos(n.*pi.*t.*(1/40)),n,1:10);
U = sum(utemp);
U = subs(U,t,0:35);
U = subs(U,x,15);
plot(0:35,U)
```

picture:
```
syms x n t
y1 = (x./50).*sin((n.*pi.*x)./120);
y2 = (120-x./70).*sin((n.*pi.*x)./120);
I1 = int(y1,x,0,50);
I2 = int(y2,x,50,120);
Cn = ((1/60).*I1)+((1/60).*I2);
utemp =
subs(Cn.*sin(n.*pi.*x.*(1/120)).*cos(n.*pi.*t.*(1/40)),n,1:10);
U = sum(utemp);
U = subs(U,t,0:35);
U = subs(U,x,25);
plot(0:35,U)
```

picture:
```
syms x n t
y1 = (x./50).*sin((n.*pi.*x)./120);
y2 = (120-x./70).*sin((n.*pi.*x)./120);
I1 = int(y1,x,0,50);
I2 = int(y2,x,50,120);
Cn = ((1/60).*I1)+((1/60).*I2);
utemp =
subs(Cn.*sin(n.*pi.*x.*(1/120)).*cos(n.*pi.*t.*(1/40)),n,1:10);
U = sum(utemp);
U = subs(U,t,0:35);
U = subs(U,x,47);
plot(0:35,U)
```

picture:
```
syms x n t
y1 = sin((n.*pi.*x)./90);
I1 = int(y1,x,10,15);
Kn = ((2.*n.*pi).*I1);
utemp = subs(Kn.*sin(n.*pi.*x.*(1/90)).*sin(n.*pi.*t.*(1/10)),n,1:10);
U = sum(utemp);
U = subs(U,t,1);
U = subs(U,x,0:90);
figure(1)
plot(0:90,U)
syms x n t
y1 = sin((n.*pi.*x)./90);
I1 = int(y1,x,10,15);
Kn = ((2.*n.*pi).*I1);
utemp = subs(Kn.*sin(n.*pi.*x.*(1/90)).*sin(n.*pi.*t.*(1/10)),n,1:10);
U = sum(utemp);
U = subs(U,t,3.5);
U = subs(U,x,0:90);
figure(1)
plot(0:90,U)
```

picture:
```
syms x n t
y1 = sin((n.*pi.*x)./90);
I1 = int(y1,x,10,15);
Kn = ((2.*n.*pi).*I1);
utemp = subs(Kn.*sin(n.*pi.*x.*(1/90)).*sin(n.*pi.*t.*(1/10)),n,1:10);
U = sum(utemp);
U = subs(U,t,11);
U = subs(U,x,0:90);
figure(1)
plot(0:90,U)
```

picture:
```
syms x n t
y1 = sin((n.*pi.*x)./90);
I1 = int(y1,x,10,15);
Kn = ((2.*n.*pi).*I1);
utemp = subs(Kn.*sin(n.*pi.*x.*(1/90)).*sin(n.*pi.*t.*(1/10)),n,1:10);
U = sum(utemp);
U = subs(U,t,18);
U = subs(U,x,0:90);
figure(1)
plot(0:90,U)
```

picture:
■ Sounds from a piano, unlike the guitar, are put into effect by striking strings. When a player presses a key, it causes a hammer to strike the strings. The corresponding IBVP is

*s*is a position of the left hammer's end and

*h*is the width of the hammer. It is assumed that both

*s*and

*s+h*are within the string length ℓ. ■