MAT4847
Assignment 3
DUE : 5 OCTOBER 2026
, Assignment 03
Chapters 4–7 (Haberman, 4th Edition)
Unique Number:
Closing Date: 05 October 2026
Question 1 — PB 4.4.2(c) (15 marks)
Problem (PB). A non-uniform string with constant coefficients and a restoring body
force Q = a u, with a < 0, satisfies
ρ 0 ut t =T 0 u x x + a u , u( 0 , t) = u( L , t) = 0 , u( x , 0) = 0 , ut ( x , 0) = f ( x ) .
Separation of variables. Let u ( x , t) = ϕ( x ) h ( t ). Substituting,
ρ0 h ″ T0ϕ ″
ρ 0 ϕ h ″ = T0 ϕ ″ h + a ϕ h⇒ −a= =−T 0 μ ,
h ϕ
which gives the spatial eigenvalue problem
ϕ ″ + μ ϕ = 0 ,(ϕ0 ) = ϕ( L ) = 0 .
This is the standard eigenvalue problem already solved in the Prescribed Book (§3.3 /
§4.4), so its result is used directly:
( )
2
nπ nπx
μn = , ϕn ( x ) = sin ,n=1,2,3,…
L L
Time equation.
√
2
2 T 0 ( n π / L) − a
ρ 0 h ″ =( a−T 0 μ n ) h ⇒h ″ + ω h = 0 , ωn =
n .
ρ0
2
Since a < 0, we have ωn > 0 for every n (the restoring force only raises the frequency), so
Assignment 3
DUE : 5 OCTOBER 2026
, Assignment 03
Chapters 4–7 (Haberman, 4th Edition)
Unique Number:
Closing Date: 05 October 2026
Question 1 — PB 4.4.2(c) (15 marks)
Problem (PB). A non-uniform string with constant coefficients and a restoring body
force Q = a u, with a < 0, satisfies
ρ 0 ut t =T 0 u x x + a u , u( 0 , t) = u( L , t) = 0 , u( x , 0) = 0 , ut ( x , 0) = f ( x ) .
Separation of variables. Let u ( x , t) = ϕ( x ) h ( t ). Substituting,
ρ0 h ″ T0ϕ ″
ρ 0 ϕ h ″ = T0 ϕ ″ h + a ϕ h⇒ −a= =−T 0 μ ,
h ϕ
which gives the spatial eigenvalue problem
ϕ ″ + μ ϕ = 0 ,(ϕ0 ) = ϕ( L ) = 0 .
This is the standard eigenvalue problem already solved in the Prescribed Book (§3.3 /
§4.4), so its result is used directly:
( )
2
nπ nπx
μn = , ϕn ( x ) = sin ,n=1,2,3,…
L L
Time equation.
√
2
2 T 0 ( n π / L) − a
ρ 0 h ″ =( a−T 0 μ n ) h ⇒h ″ + ω h = 0 , ωn =
n .
ρ0
2
Since a < 0, we have ωn > 0 for every n (the restoring force only raises the frequency), so
