Question Number 218539 by SdC355 last updated on 11/Apr/25
$${S}\mathrm{olve} \\ $$$$\frac{\partial^{\mathrm{2}} {w}}{\partial{t}^{\mathrm{2}} }={c}^{\mathrm{2}} \frac{\partial^{\mathrm{2}} {w}}{\partial{x}^{\mathrm{2}} } \\ $$$${w}\left(\mathrm{0},{t}\right)={f}\left({t}\right)\:,\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{w}\left({x},{t}\right)=\mathrm{0}\:\left(\mathrm{Boundary}\:\mathrm{Condition}\right) \\ $$$${w}\left({x},\mathrm{0}\right)=\mathrm{0}\:,\:{w}_{{t}} \left({x},\mathrm{0}\right)=\mathrm{0}\:\left(\mathrm{Initial}\:\mathrm{Condition}\right) \\ $$$${f}\left({t}\right)\begin{cases}{\mathrm{sin}\left({t}\right)\:,\:{t}\in\left[\mathrm{0},\mathrm{2}\pi\right)}\\{\mathrm{0}\:,\:\mathrm{otherwise}}\end{cases} \\ $$
Answered by MrGaster last updated on 11/Apr/25
![(∂^2 w/∂t^2 )=c^2 (∂^2 w/∂x^2 ) L{w}≜W(x,s) s^2 W=^(∵w(x,0)=0∧w_t (x,0)=0) c^2 (d^2 W/dx^2 ) W(x,s)=A(s)e^(−sx/c) +B(s)e^(sx/c) lim_(x→∞) W(x,s)=0⇒B(s)=0 W(0,s)=L{f(t)}=F(s)⇒A(s)=F(s) W(x,s)=F(s)e^(−sx/c) L^(−1) {e^(−sx/c) }=δ(t−x/c) w(x,t)=f(t−x/c)H(t−x/c) f(t−x/c)= { ((sin(t−x/c),(x/c)≤t<(x/c)+2π)),((0 ,otherwise )) :} w(x,t)=sin(t−(x/c))H(t−(x/c))[H(t−(x/c))−H(t−(x/c)−2π)] determinant (((w(x,t)=sin(t−(x/c))H(t−(x/c))H(2π+(x/c)−t))))](https://www.tinkutara.com/question/Q218541.png)
$$\frac{\partial^{\mathrm{2}} {w}}{\partial{t}^{\mathrm{2}} }={c}^{\mathrm{2}} \frac{\partial^{\mathrm{2}} {w}}{\partial{x}^{\mathrm{2}} } \\ $$$$\mathcal{L}\left\{{w}\right\}\triangleq{W}\left({x},{s}\right) \\ $$$${s}^{\mathrm{2}} {W}\overset{\because{w}\left({x},\mathrm{0}\right)=\mathrm{0}\wedge{w}_{{t}} \left({x},\mathrm{0}\right)=\mathrm{0}} {=}{c}^{\mathrm{2}} \frac{{d}^{\mathrm{2}} {W}}{{dx}^{\mathrm{2}} } \\ $$$${W}\left({x},{s}\right)={A}\left({s}\right){e}^{−{sx}/{c}} +{B}\left({s}\right){e}^{{sx}/{c}} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}{W}\left({x},{s}\right)=\mathrm{0}\Rightarrow{B}\left({s}\right)=\mathrm{0} \\ $$$${W}\left(\mathrm{0},{s}\right)=\mathcal{L}\left\{{f}\left({t}\right)\right\}={F}\left({s}\right)\Rightarrow{A}\left({s}\right)={F}\left({s}\right) \\ $$$${W}\left({x},{s}\right)={F}\left({s}\right){e}^{−{sx}/{c}} \\ $$$$\mathcal{L}^{−\mathrm{1}} \left\{{e}^{−{sx}/{c}} \right\}=\delta\left({t}−{x}/{c}\right) \\ $$$${w}\left({x},{t}\right)={f}\left({t}−{x}/{c}\right){H}\left({t}−{x}/{c}\right) \\ $$$${f}\left({t}−{x}/{c}\right)=\begin{cases}{\mathrm{sin}\left({t}−{x}/{c}\right),\frac{{x}}{{c}}\leq{t}<\frac{{x}}{{c}}+\mathrm{2}\pi}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:,\mathrm{otherwise}\:}\end{cases} \\ $$$${w}\left({x},{t}\right)=\mathrm{sin}\left({t}−\frac{{x}}{{c}}\right){H}\left({t}−\frac{{x}}{{c}}\right)\left[{H}\left({t}−\frac{{x}}{{c}}\right)−{H}\left({t}−\frac{{x}}{{c}}−\mathrm{2}\pi\right)\right] \\ $$$$\begin{array}{|c|}{{w}\left({x},{t}\right)=\mathrm{sin}\left({t}−\frac{{x}}{{c}}\right){H}\left({t}−\frac{{x}}{{c}}\right){H}\left(\mathrm{2}\pi+\frac{{x}}{{c}}−{t}\right)}\\\hline\end{array} \\ $$