One Dimensional Wave Equation is (A) 𝛿2y/𝛿t2 = α2 𝛿2y/𝛿x2 (B) 𝛿y/𝛿t = α2 𝛿2y/𝛿x2 (C) 𝛿2y/𝛿t2 = -𝛿2y/𝛿x2 (D) None of these
By BYJU'S Exam Prep
Updated on: October 17th, 2023
One dimensional wave equation is 𝛿2y/𝛿t2 = α2 𝛿2y/𝛿x2. To understand this, you need to understand about Wave equation first. A second-order linear partial differential equation is used to describe waves in the wave equation (like mechanical waves); it is known as the Wave equation.
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The Partial Differential equation is given as,
A 𝛿2u/𝛿x2 + B 𝛿2u/𝛿x𝛿y + C 𝛿2u/𝛿y2 + D 𝛿y/𝛿x + E 𝛿u/𝛿y = F
For the One-Dimensional equation,
α2 𝛿2y/𝛿x2 = 𝛿2y/𝛿t2
where, A = α , B = 0, C = -1
Put all the values in equation (1)
∴ 0 – 4 (α )(-1)
4α > 0.
So, this is a one-dimensional wave equation.
- 𝛿y/𝛿t = α2 𝛿2y/𝛿x2
Having A = α2, B = 0 and C = 0
Put all the values in equation (1), we get
0 – 4(α )(0) = 0, therefore it shows parabolic function.
So, this is a one-dimensional heat equation.
- 𝛿2y/𝛿t2 = -𝛿2y/𝛿x2
having A = 1, B = 0, C = 1
Put all the values in equation (1), we get
0 – 4(1)(1) = -4, therefore it shows elliptical function.
So, this is a two-dimensional heat equation.
Summary:
One Dimensional Wave Equation is (A) 𝛿2y/𝛿t2 = α2 𝛿2y/𝛿x2 (B) 𝛿y/𝛿t = α2 𝛿2y/𝛿x2 (C) 𝛿2y/𝛿t2 = -𝛿2y/𝛿x2 (D) None of these
𝛿2y/𝛿t2 = α2 𝛿2y/𝛿x2 is one-dimensional wave equation. A linear second-order partial differential equation that describes the propagation of oscillations at a fixed speed in some quantity.
