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UDC 517.947.44 : 517.946.9
QUASI-PERIODIC AND PERIODIC SOLUTIONS
OF PROBLEMS WITH MOVING BOUNDARIES
FOR THE WAVE EQUATION
IN ONE-DIMENSIONAL SPACE
V. I. KVALVASSER, Yu. P. SAMARIN
In works [1, 2], exact solutions were constructed for the problem of oscillations of a segment of a string with uniformly moving boundaries. In doing so, use was made of the fact that the configurations of the string at times forming a geometric progression with denominator \(a>1\) differ only by the choice of scale for the \(x\)-axis; more precisely: solutions of the wave equation
\[ \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}, \tag{1} \]
corresponding to such a problem, are subject to the condition
\[ u(ax,at)=u(x,t), \tag{2} \]
called below the quasi-periodicity condition.
However, it turns out that uniformity of the motion of the boundaries is not necessary for the fulfillment of relation (2), which defines a certain class of solutions of problems with moving boundaries.
Let two points \((0,t_0)\) and \((0,at_0)\) be joined by some curve \(x=f(t)\), representing the motion of a boundary with variable subsonic velocity, i.e. \(|f'(t)|<1\) (Fig. 1). For this curve we introduce into consideration the conjugate curve \(1—2\), representing the motion of the second boundary; here the conjugate curve is, by definition, chosen so that the solution of the corresponding mixed problem for equation (1), with arbitrary initial conditions at \(t=t_0\) and zero boundary conditions of the first kind, satisfies the quasi-periodicity condition (2) at \(t=t_0\). For this it is necessary and sufficient that the signals issuing from an arbitrary point 3, located on the line \(t=t_0\), arrive at the point 4, located on the continuation of the line \(0—3\) at \(t=at_0\), having the previous directions—
Fig. 1 Fig. 2
of the boundary; here the conjugate curve, by definition, is chosen so that the solution of the corresponding mixed problem for equation (1), with arbitrary initial conditions at \(t=t_0\) and zero boundary conditions of the first kind, satisfies the quasi-periodicity condition (2) at \(t=t_0\). For this it is necessary and sufficient that the signals issuing from an arbitrary point 3, located on the line \(t=t_0\), arrive at the point 4, located on the continuation of the line \(0—3\) at \(t=at_0\), having the previous directions-
propagation. As is seen from Fig. 1, this makes it possible to determine two points 5 and 6 belonging to the conjugate curve 1—2. It is not difficult to show that its equation is established by the relations:
\[ t_5=\frac{a-1}{2a}\,x_8+\frac{a+1}{2a}\,t_8,\qquad x_5=\frac{a+1}{2a}\,x_8+\frac{a-1}{2a}\,t_8; \]
\[ t_6=\frac{a-1}{2}\,x_7+\frac{a+1}{2}\,t_7,\qquad x_6=\frac{a+1}{2}\,x_7+\frac{a-1}{2}\,t_7 \]
(the indices correspond to the designations of the points in Fig. 1), whence it is clear that the conjugate curve \(x=f_*(t)\) is obtained by an affine transformation of the given one (for a straight line the conjugate is also a straight line—the case considered in the cited works).
The curvilinear trapezoid obtained in this way can be subjected to a homothety with center at the point \(O\). As a result, in each of the trapezoids obtained the wave process will occur in the same way, with accuracy up to the choice of scale along the axes \(x\) and \(t\), so that (2) is fulfilled for all \(t\geqslant t_0\), and one can construct the exact solution of the mixed problem for equation (1):
\[ u\big|_{t=t_0}=\varphi(x),\qquad \frac{\partial u}{\partial t}\bigg|_{t=t_0}=\psi(x),\quad x\in[0,x_0]; \tag{3} \]
\[ u\big|_{x=f(t)}=0,\qquad u\big|_{x=f_*(t)}=0,\quad t\geqslant t_0. \tag{4} \]
The equations \(x=f(t)\) and \(x=f_*(t)\), determining the laws of motion of the boundaries, are written in the previous form, although in (4) they are considered prescribed for arbitrary \(t\geqslant t_0\).
The general solution of (1) is \(u=\xi(x-t)+\zeta(x+t)\), and the condition (4) at \(x=f(t)\) leads to the equality \(\xi(f(t)-t)+\zeta(f(t)+t)=0\), from which one can find that
\[ \xi(\tau)=-\zeta(\tau+2\chi(\tau)), \]
where by \(\tau\) is denoted the difference \(f(t)-t\), and the function \(\chi(\tau)\) is introduced as the solution of the equation \(\tau=f(t)-t\) with respect to \(t\); therefore
\[ u=\zeta(x+t)-\zeta(x-t+2\chi(x-t)). \tag{5} \]
Since \(f(at)=af(t)\) and \(\chi(a\tau)=a\chi(\tau)\), it follows from (2) that \(\zeta(a\tau)=\zeta(\tau)\), which holds for
\[ \zeta(\tau)=\sum_n c_n\exp(ink\ln\tau),\qquad k=\frac{2\pi}{\ln a},\quad n=\pm1,\ \pm2,\ldots \]
Using this, one can write (5) as
\[ u=\sum_n c_n\{\exp[ink\ln(x+t)]-\exp[ink\ln\nu(x-t)]\},\qquad \nu(\tau)=\tau+2\chi(\tau). \tag{6} \]
If (6) is substituted into (3),
\[ \sum_n c_n\{\exp[ink\ln(x+t_0)]-\exp[ink\ln\nu(x-t_0)]\}=\varphi(x), \]
\[ ik\sum_n nc_n\left\{ \frac{\exp[ink\ln(x+t_0)]}{x+t_0} + \frac{\exp[ink\ln\nu(x-t_0)]\,\nu'(x-t_0)}{\nu(x-t_0)} \right\}=\psi(x) \]
and the last equality is integrated, then it is not difficult to find that
\[ \int_0^x \psi(\eta)\,d\eta+\varphi(x) = 2\sum_n c_n\{\exp[ink\ln(x+t_0)]-\exp(ink\ln t_0)\}, \tag{7} \]
\[ \int_0^x \psi(\eta)\,d\eta-\varphi(x) = 2\sum_n c_n\{\exp[ink\ln\nu(x-t_0)]-\exp(ink\ln t_0)\}. \tag{8} \]
After the change of variables, putting \(x=f(t)+t-t_0\) in (7) and \(x=f(t)-t+t_0\) in (8), it turns out that (8) will be a consequence of (7) if and only if
\[ \varphi\bigl(f(t)+t-t_0\bigr)+\psi\bigl(f(t)-t+t_0\bigr)+ \int_{f(t)-t+t_0}^{f(t)+t-t_0}\psi(\eta)\,d\eta=0. \tag{9} \]
With the aid of (9), the functions \(\varphi(x)\) and \(\psi(x)\) can be continued for
\[ x\in[-(a t_0-t_0-a x_0),\,0] \]
as follows:
\[ \varphi(x)=-\varphi\bigl(x+2\chi(x-t_0)-2t_0\bigr), \tag{10} \]
\[ \psi(x)=\bigl[1+2\chi'(x-t_0)\bigr]\psi\bigl(x+2\chi(x-t_0)-2t_0\bigr). \]
For \(x\in[-(a-1)t_0,\,-(a t_0-t_0-a x_0)]\), \(\varphi(x)\) and \(\psi(x)\) are continued “quasi-periodically,” so that condition (4) for \(x=f_*(t)\) is satisfied:
\[ \varphi(x)=\varphi\left(\frac{x-t_0}{a}+t_0\right),\qquad \psi(x)=\frac{1}{a}\psi\left(\frac{x-t_0}{a}+t_0\right). \tag{11} \]
With the aid of (3), (10), and (11), the coefficients \(c_n\) from (7) are determined:
\[ c_n=\frac{1}{2\ln a} \int_{-\frac{a t_0-x_0-t_0}{a}}^{x_0} \left[ \varphi(x)+\int_0^x \psi(\eta)\,d\eta \right] \exp[-i n k\ln(x+t_0)]\,\frac{dx}{x+t_0}. \]
In the preceding arguments it was assumed that the prescribed curve \(x=f(t)\) passes through two points lying on the \(t\)-axis, which, of course, is not essential, since, by applying the corresponding Lorentz transformation, one can always map any “subsonic” line onto a line parallel to the \(t\)-axis.
Further, in Fig. 1 the case is depicted in which, on the path from point 3 to point 4, the signal undergoes one reflection from each boundary. There may be more such reflections; in this case some part of the conjugate curve will be arbitrary. This does not substantially affect the construction of the solution.
Following this method, one can construct other classes of solutions of problems with moving boundaries for equation (1), determined by one or another functional equation. Consider, for example, periodic solutions (Fig. 2), for which
\[ u(x,t+T)=u(x,t). \tag{12} \]
The periodicity condition (12) is characteristic for the solution of a mixed problem with fixed boundaries. However, as is seen from Fig. 2, the condition that the boundary be fixed is not necessary, and one can consider the problem (3), (4), where \(f_*(t)=f(t+T/2)+T/2\). Analogously to the preceding case, one obtains (5), where, in view of (12), \(\zeta(\tau+T)=\zeta(\tau)\), i.e., one may assume that
\[ \zeta(\tau)=\sum_n c_n\exp\left(\frac{2\pi i n}{T}\tau\right). \]
Thus,
\[ u=\sum_n c_n\left\{ \exp\left[\frac{2\pi i n}{T}(x+t)\right] - \exp\left[\frac{2\pi i n}{T}\nu(x-t)\right] \right\}. \]
For \(x\in[-(T-x_0),\,0]\), (10) holds with \(t_0=0\), and therefore the unknown coefficients can be found:
\[ c_n=\frac{1}{2T} \int_{-(T-x_0)}^{x_0} \left[ \varphi(x)+\int_0^x \psi(\eta)\,d\eta \right] \exp\left(-\frac{2\pi i n}{T}x\right)\,dx. \]
References
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Nikolai E. L. On transverse vibrations of a segment of a string whose length varies. Proceedings on Mechanics. Moscow, 1955.
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Balazs N. L. J. of Math. Analysis and Appl. 3, No. 3, 1961, 472–484. New York.
Received by the editors March 22, 1965.
Kuibyshev Polytechnic Institute