MATHEMATICS - 2
PROJECT BASED LEARNING
Group Members:
Sumit Verma – 21103163
Twarit Agarwal – 21103171
Viraaj Singh Luther – 21103144
Sanskar Khandelwal – 21103172
STRING EQUATION
 INTRODUCTION TO PARTIAL DIFFERENTIAL
EQUATIONS:
Partial Differential are often used to construct models of the
most theories underlying physics and engineering. For example,
the system of partial differential equations known as Maxwell’s
equation can derive the entire theory of electricity and
magnetism from these equations.
Our Goal here is to develop the most basic ideas from the
theory of partial differential equations, and apply them to the
wave equation. In particular, we will present some of the
elegant mathematics used to describe the nature of a string.
 HISTORY:
Partial differential equation arose in the context of the
development of models in the physics of continuous media,
example: Vibrating strings, electrostatics, fluid flow and later
theories of heat conduction.
The one-dimensional wave equation:
∂2u/∂2t = ∂2u/∂2x
was introduced and analysed by D’ Alembert in 1752.
A central connection between PDE and the mainstream of
Mathematical development of 19th century arose from the role
of PDE in theory of analytic functions of a complex variable.
Cauchy Riemann equation:
∂u/∂x = ∂v/∂y
- (1)
∂u/∂y = -∂v/∂x
- (2)
Later, this became the central defining feature of analytic
function.
Types Of Second Order PDEsThere are 3-types of second-order PDEs in mechanics.
Elliptic PDE b2−4ac<0
 Eg- Describing the equation of elasticity.
Hyperbolic PDE b2−4ac>0
 Eg- Wave equation
Parabolic PDE b2−4ac=0
 Eg- Heat equation
INTRODUCTION TO WAVE EQUATION:
In the mathematical sense, a wave is any function that moves to
displace any function f(x) to the right, just change its arguments from
x to x to x-x0, where x0>0.If we let x0=vt where v>0 and time is, then
the displacement increases with increasing time. So, f(x-vt)
represents a leftward, or backward propagating wave where V is the
velocity of wave.
The wave equation is a linear partial differential equation for the
description of waves or standing wave fields as they occur in classical
physics- such as mechanical waves (eg- water waves, sound waves
and seismic waves) or electromagnetic waves (including light waves).
It arises in fields like acoustics, electromagnetism, and fluid
dynamics. A single wave propagating in a pre-defined direction can
also be described with the one-way wave equation
The following is one-dimensional wave equations:
𝝏𝟐 𝒖
𝝏𝟐 𝒕
𝟐
2𝝏 𝒖
=c
𝝏𝟐 𝒙
Derivation of equation of vibrating string:
Let us consider a small traverse vibration of an elastic string of length
l, which is stretched and then fixed at two ends. We will discuss the
situation under the no external forces acting on It.
Take an end of the string as origin & the string is in equilibrium
position as the x-axis & the line through the origin and perpendicular
to the x-axis as the Y-axis.
We make the following assumptions
1) The motion takes pose entirely in one plane. This plane is chosen
as x-y plane.
2) In this plane, each particle of the string moves in a direction
perpendicular to the equilibrium position of the string
3) The tension T is very large compared with the weight of the string
and hence the gravitational force may be neglected
(4) The effect of friction in negligible.
(5) The slope of the deflection curve is small at all points
(6) The string is perfectly flexible. It can transmit only tension but not
bending or shearing forces.
Since displacement is only vertical.
Y = u(v, t)
 Apply Newton’s law to an infinitesimally small segment of string
between x & x+Δx.
 Assume string has mass density f0(x)
Mass = ρ0(x). Δx
dy/dx = ∂u/∂x = tan(θ(x, t))
Since, there is negligible gravitational force and force due to
resistance, hence force is only due to tension.
F = ma ; ρ0(x). Δx. ∂2u/∂2t = T(x+Δx,t)sin(θ(x+Δx,t))
-T(x, t)sin(θ(x,t))
Dividing by Δx, taking the limit as Δx -> 0
ρ0(x) ∂2u/∂2t = ∂(T(x,t) ∂u/∂x)
If the string is perfectly elastic:
T(x,t) = T0 (constant)
ρ0(x) ∂2u/∂2t = T0 ∂2u/∂2x
Since density is constant:
∂2u/∂2t = c2 ∂2u/∂2x
Where,
C2 = T0/ ρ0
Boundary Conditions:
At 0, x = 0 and y = 0; dy/dt = 0 as t = 0;
At A, x = 1 and y = 0;
Solutions of one dimension wave equation:
Let u = X(x)T(t)
Then, XT’’ = c2X’’T
X’’/X = 1/c2(T’’/T) = k (let)
X’’ – kX = 0
&
T’’ – kc2T = 0
Case 1 -> When k=0
X’’ = 0 & T’’ = 0
∴ u(x,t) = (c1x + c2)(c3t + c4)
Case 2 -> When k = m2 (m ≠ 0)
X’’ – m2x = 0
X = (c1emx + c2e-mx)
T’’ – m2c2 = 0
∴
T = (c3emct + c4e-mct)
u(x,t) = [c1emx + c2e-mx] [c3emct + c4e-mct]
Case 3 -> When k = -m2 (m ≠ 0)
X’’ + m2x = 0 => X = [c1cosmx + c2sinmx]
T’’ + m2c2T = 0 => T = [c3cosmct + c4sinmct]
∴ u(x,t) = [c1cosmx + c2sinmx] [c3cosmct + c4sinmct]
Since the string is vibrating, the solution must contain trigonometric
terms. So, the solution by case 3 will be the best solution.
Using homogeneous boundary conditions ->
u(0,t) = 0 = u(l,t)
 m = nπ/l
& c1 = 0
Hence,
u(x,t) = Σ[Ancos(nπct/l) + Bnsin(nπct/l)sin(nπx/l)]
If the initial conditions ->
u(x,0) = f(x)
ut(x,0) = g(x)
 f(x) = ΣAnsin nπx/l
𝑙
=> An = 2/l ∫0 𝑓(𝑥 ) sin (
𝑛πx
l
) dx
 g(x) = ΣBn (nπc/l) sin (nπx/l)
l
𝑛πx
=> Bn = 2/nπc ∫𝑛=1 𝑔(𝑥 ) sin (
l
) dx
PHYSICAL INTERPRETATION:
Model for vibrating string:
u(x,t) = Σ[Ancos(nπct/l) + Bnsin(nπct/l)sin(nπx/l)]
 Musical instruments
 Each value of n gives a normal mode of vibration
 Intensity depends on the amplitude
 Time dependence is simple harmonic with circular frequency,
𝑛πc
, which is the number of oscillations of 2π units of time
l
 The sound produced consists of superposition of the infinite
number of natural frequencies, n =1,2,…
 The normal mode, n = 1, is called the first harmonic or
fundamental mode
πc
 This mode has circular frequency,
l
 Higher natural frequencies have higher pitch
 Fundamental frequency varied by changing, c =
o Tune by changing tension, T0
o Different ρ0 for different strings (range of notes)
o Musician varies pitch by varying the length L (clamping
string)
 Higher harmonics for stringed instruments are all integral
multiples (pleasing to the ear)