MATHEMATICAL PHYSICS

B. G. KORENEV

HEAT WAVES IN A THIN UNBOUNDED PLATE LYING ON A CONTINUOUS HOMOGENEOUS FOUNDATION

(Presented by Academician S. L. Sobolev on 13 VII 1956)

In this note we consider problems on heat waves in a thin plate, whose thickness \(h\) is small, and therefore it is assumed that the temperature does not vary through the thickness; the specific heat \(c\), density \(\rho\), and thermal-conductivity coefficient \(\lambda\) are constant; the coordinate axes \(x, y\) are placed in the middle plane. Suppose that boundary conditions of the third kind are prescribed on the upper plane, while the lower plane is in close contact with a continuous and homogeneous foundation in planes parallel to \(Oxy\). This problem has much in common with problems on the bending of plates on a continuous elastic foundation \((^1)\) and is closely connected with problems of the theory of heat conduction considered in \((^{2,3})\).

We first consider an auxiliary problem. Let heat sources
\(q_3 = A_1 \sin \omega t \sin \alpha x \sin \beta y + A_2 \cos \omega t \sin \alpha x \sin \beta y\) be applied to the unbounded plate. We denote the heat flux leaving through the upper plane by \(q_1\), and through the lower plane by
\(q_2 = (A_5 \sin \omega t + A_6 \cos \omega t)\sin \alpha x \sin \beta y\). The temperature of the plate is
\(T = (A_3 \sin \omega t + A_4 \cos \omega t)\sin \alpha x \sin \beta y\), \(q_1=\delta T\). Substituting their expressions for \(q_1, q_2, q_3, T\) into the heat-conduction equation and equating the coefficients of \(\sin \omega t\) and \(\cos \omega t\), we obtain

\[ \begin{aligned} -\lambda h(\alpha^2+\beta^2)A_3 &= -c\omega A_4 - A_1 + \delta A_3 + A_5,\\ -\lambda h(\alpha^2+\beta^2)A_4 &= c\omega A_3 - A_2 + \delta A_4 + A_6 . \end{aligned} \tag{1} \]

Between the coefficients \(A_5, A_6\) and \(A_3, A_4\) there exist relations determined by the properties of the homogeneous continuous foundation.

Suppose that the properties of the foundation are described by kernels \(K_1(r), K_2(r)\), such that, under the action of a unit source varying according to the law \(\sin \omega t\), the temperature of the upper surface of the foundation varies according to the law
\(T_1 = K_1(r)\sin \omega t + K_2(r)\cos \omega t\). Therefore, if we have a source \(\cos \omega t\), then the temperature of the upper surface of the foundation varies according to the law

\[ T_1 = K_1(r)\cos \omega t - K_2(r)\sin \omega t . \]

From this it is easy to determine the dependences between \(A_3, A_4\) and \(A_5, A_6\). Indeed, if the sources \(A_5\sin \omega t \sin \alpha x \sin \beta y\) act on the foundation, then

\[ T^{(1)} = P_1(\alpha,\beta)\sin \omega t + P_2(\alpha,\beta)\cos \omega t, \tag{2} \]

where

\[ \begin{aligned} P_1(\alpha,\beta) &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} K_1\left(\sqrt{x^2+y^2}\right)\sin \alpha x \sin \beta y\,dx\,dy,\\ P_2(\alpha,\beta) &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} K_2\left(\sqrt{x^2+y^2}\right)\sin \alpha x \sin \beta y\,dx\,dy; \end{aligned} \tag{3} \]

in our case \(P_1(\alpha,\beta)=P_1(\gamma)\), \(P_2(\alpha,\beta)=P_2(\gamma)\), \(\gamma=\sqrt{\alpha^2+\beta^2}\), therefore

\[ T^{(1)}=A_5\,[P_1(\gamma)\sin \omega t+P_2(\gamma)\cos \omega t]. \tag{4} \]

If heat sources \(A_6\sin \omega t \sin \alpha x \sin \beta y\) are applied to the surface of the foundation, then

\[ T^{(2)}=A_6\,[P_1(\gamma)\cos \omega t-P_2(\gamma)\sin \omega t]. \tag{5} \]

For brevity of notation, introduce the designations

\[ P_1[\lambda h\gamma^2+\delta]-c\omega P_2+1=\Phi_1,\qquad P_2[\lambda h\gamma^2+\delta]+c\omega P_1=\Phi_2. \tag{6} \]

Then

\[ \begin{gathered} A_5=\frac{A_1\Phi_1+A_2\Phi_2}{\Phi_1^2+\Phi_2^2},\qquad A_6=\frac{-A_1\Phi_2+A_2\Phi_1}{\Phi_1^2+\Phi_2^2},\\ A_3=A_1\frac{\Phi_1P_1+\Phi_2P_2}{\Phi_1^2+\Phi_2^2} +A_2\frac{\Phi_2P_1-\Phi_1P_2}{\Phi_1^2+\Phi_2^2},\\ A_4=A_1\frac{\Phi_1P_2+\Phi_2P_1}{\Phi_1^2+\Phi_2^2} +A_2\frac{\Phi_2P_2-\Phi_1P_1}{\Phi_1^2+\Phi_2^2}. \end{gathered} \tag{7} \]

It is not difficult to verify that, in essence without loss of generality, one may restrict oneself to the consideration of the case \(A_1\ne0\), \(A_2=0\); then the sought function \(T\) will have the form

\[ T=\frac{A_1\sin \alpha x \sin \beta y}{\Phi_1^2+\Phi_2^2} \{[\Phi_1P_1+\Phi_2P_2]\sin \omega t+[\Phi_1P_2+\Phi_2P_1]\cos \omega t\}; \tag{8} \]

if \(q_3=\sum\sum A_{1mn}\sin \alpha_m x\sin \beta_n y\), then

\[ T=\sum\sum \frac{A_{1mn}\sin \alpha_m x\sin \beta_n y}{\Phi_{1mn}^2+\Phi_{2mn}^2} \{[\Phi_1P_1+\Phi_2P_2]\sin \omega t+ [\Phi_1P_2+\Phi_2P_1]\cos \omega t\}. \tag{9} \]

If a point heat source \(Q\sin \omega t\) acts on the plate, then

\[ T=\frac{Q}{\pi^2} \left\{ \sin \omega t\int_0^\infty\int_0^\infty \frac{\cos \alpha x\cos \beta y\,d\alpha\,d\beta}{\widetilde{\Phi}_1(\alpha,\beta)} +\cos \omega t\int_0^\infty\int_0^\infty \frac{\cos \alpha x\cos \beta y\,d\alpha\,d\beta}{\widetilde{\Phi}_2(\alpha,\beta)} \right\}, \tag{10} \]

where

\[ \widetilde{\Phi}_1(\alpha,\beta)= \frac{\Phi_1^2+\Phi_2^2}{\Phi_1P_1+\Phi_2P_2},\qquad \widetilde{\Phi}_2(\alpha,\beta)= \frac{\Phi_1^2+\Phi_2^2}{\Phi_1P_2+\Phi_2P_1}. \]

This formula can be written in the form

\[ T=\frac{Q}{2\pi} \left[ \sin \omega t\int_0^\infty \frac{\gamma J_0(\gamma r)\,d\gamma}{\widetilde{\Phi}_1(\gamma)} +\cos \omega t\int_0^\infty \frac{\gamma J_0(\gamma r)\,d\gamma}{\widetilde{\Phi}_2(\gamma)} \right]. \tag{11} \]

Using the results of (1), one can easily analyze a number of other problems concerning the action of heat sources on a plate and on a rod.

To determine the functions \(P_1(\alpha,\beta)\), \(P_2(\alpha,\beta)\) in the case where the foundation is a homogeneous half-space, it is necessary to consider the problem of heat waves in a half-space with sources \(\sin \alpha x \sin \beta y \sin \omega t\) on the boundary. This problem is equivalent to the problem of bending of a semi-infinite beam with two elastic characteristics, loaded at the end.

Received
19 IV 1956

CITED LITERATURE

\(^{1}\) B. G. Korenev, Problems in the Calculation of Beams and Plates on an Elastic Foundation, Moscow, 1954.
\(^{2}\) B. G. Korenev, DAN, 107, No. 2 (1956).
\(^{3}\) B. G. Korenev, DAN, 112, No. 1 (1957).