Question and Answers Forum

All Questions      Topic List

Mechanics Questions

Previous in All Question      Next in All Question      

Previous in Mechanics      Next in Mechanics      

Question Number 196785 by Spillover last updated on 31/Aug/23

Answered by aleks041103 last updated on 03/Oct/23

Separation of variables Ψ(x,t)=X(x)T(t) ⇒X′′T=−(1/c^2 )XT′′ ⇒((X′′)/X)=−(1/c^2 ) ((T ′′)/T)=−a^2 ⇒X′′+a^2 X=0 ⇒X=sin(ax+b) T ′′ −(ac)^2 T=0⇒T=sinh(act+d) ⇒XT=sin(ax+b)sinh(act+d) X(0)=X(L)=0 ⇒sin(b)=sin(aL+b)=0 ⇒b=0, aL=πk⇒a=((kπ)/L),k∈Z ⇒Ψ(x,t)=Σ_(k=−∞) ^∞ A_k sin(((πkx)/L))sinh(((πkct)/L)+B_k ) where A_k , B_k are constants determined by initial conditions.

SeparationofvariablesΨ(x,t)=X(x)T(t)⇒X″T=−1c2XT″⇒X″X=−1c2T″T=−a2⇒X″+a2X=0⇒X=sin(ax+b)T″−(ac)2T=0⇒T=sinh(act+d)⇒XT=sin(ax+b)sinh(act+d)X(0)=X(L)=0⇒sin(b)=sin(aL+b)=0⇒b=0,aL=πk⇒a=kπL,k∈Z⇒Ψ(x,t)=∑∞k=−∞Aksin(πkxL)sinh(πkctL+Bk)whereAk,Bkareconstantsdeterminedbyinitialconditions.

Terms of Service

Privacy Policy

Contact: info@tinkutara.com