Complex Variables and Partial Differential Equations (3130005)

Winter-2019

Question-5c-OR

BE | Semester-3 Winter-2019 | 26-11-2019

Q5) (c)

Derive the one dimensional wave equation that governs small vibration of an elastic string. Also state physical assumptions that you make for the system.

→ We begin by studying the one-dimensional wave equation, which describe the transverse vibrations of a string. Consider the small vibrations of a string that is fastened at each end (see, Figure). We now make the following assumptions:

The string is made of a homogeneous material.
There is no effect of gravity and external forces.
The vibration takes place in a plane.

→ The mathematical model equation under these assumptions describe small vibrations of the string. Let the forces acting on a small portion

PQ

of the string. Since the string does not offer resistance to bending, the tension is tangential to the curve of the string at each point.

→ Let

T_{1}

and

T_{2}

, respectively, be the tensions at the end points

P

and

Q

. Since there is no motion in horizontal direction, the horizontal components of the tension must be constant. From the Figure, we obtain

T_{1} \cos α = T_{2} \cos β = T = constant - - - - - (1)

→ Let

- T_{1} \sin α

and

T_{2} \sin β

be two components of

T_{1}

and

T_{2}

, respectively in the vertical direction. The minus sign indicates that component at

P

is directed downward. By Newton’s second law, the resultant of these two forces is equal to the mass

ρ Δx

of the portion times the acceleration

u_{tt}

, evaluated at some point between

x

and

x + Δx

→ If

ρ

is the mass of the undeflected string per unit length and

Δx

is length of the portion of the undeflected string then we have

T_{2} \sin β - T_{1} \sin α = ρ Δx u_{tt}

\Rightarrow \frac{T_{2} \sin β}{T} - \frac{T_{1} \sin α}{T} = \frac{ρ Δx u_{tt}}{T}

\Rightarrow \frac{T_{2} \sin β}{T_{2} \cos β} - \frac{T_{1} \sin α}{T_{1} \cos α} = \frac{ρ Δx}{T} u_{tt} (∵ [1])

\Rightarrow \tan β - \tan α = \frac{ρ Δx}{T} u_{tt}

\Rightarrow \frac{1}{Δx} [u_{x} (x + Δx, t) - u_{x} (x, t)] = \frac{ρ}{T} u_{tt}

\Rightarrow u_{xx} = \frac{ρ}{T} u_{tt} (taking ∆ x ⇢ 0)

\Rightarrow u_{tt} = \frac{T}{ρ} u_{xx} \Rightarrow u_{tt} = c^{2} u_{xx} (Wave equation)

Complex Variables and Partial Differential Equations (3130005)

Q5) (c)

Derive the one dimensional wave equation that governs small vibration of an elastic string. Also state physical assumptions that you make for the system.

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