difference in elastic waves in rod and a string
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Consider a thin uniform elastic rod of length  and cross-sectional area  . Let us examine the longitudinal oscillations of such a rod. These oscillations are often, somewhat loosely, referred to as sound waves. It is again convenient to let  denote position along the rod. Thus, in equilibrium, the two ends of the rod lie at  and  . Suppose that a longitudinal wave causes an  -directed displacement  of the various elements of the rod from their equilibrium positions. Consider a thin section of the rod, of length  , lying between  and  . The displacements of the left and right boundaries of the section are  and , respectively. Thus, the change in length of the section, due to the action of the wave, is  . Now, strain in an elastic rod is defined as change in length over unperturbed length (Love 1944). Thus, the strain in the section of the rod under consideration is
(302)
In the limit  , this becomes
(303)
It is assumed that the strain is small: that is,  . Stress,  , in an elastic rod is defined as the elastic force per unit cross-sectional area (ibid.). In a conventional elastic material, the relationship between stress and strain (for relatively small strains) takes the simple form
(304)
Here,  is a constant, with the dimensions of pressure, which is known as the Young's modulus (ibid.). If the strain in a given element is positive then the stress acts to lengthen the element, and vice versa. (Similarly, in the spring-coupled mass system investigated in the preceding section, the external forces exerted on an individual spring act to lengthen it when its extension is positive, and vice versa.)
Consider the motion of a thin section of the rod lying between  and  . If  is the mass density of the rod then the section's mass is  . The stress acting on the left boundary of the section is  . Since stress is force per unit area, the force acting on the left boundary is . This force is directed in the minus  -direction, assuming that the strain is positive (i.e., the force acts to lengthen the section). Likewise, the force acting on the right boundary of the section is  , and is directed in the positive  -direction, assuming that the strain is positive (i.e., the force again acts to lengt
(302)
In the limit  , this becomes
(303)
It is assumed that the strain is small: that is,  . Stress,  , in an elastic rod is defined as the elastic force per unit cross-sectional area (ibid.). In a conventional elastic material, the relationship between stress and strain (for relatively small strains) takes the simple form
(304)
Here,  is a constant, with the dimensions of pressure, which is known as the Young's modulus (ibid.). If the strain in a given element is positive then the stress acts to lengthen the element, and vice versa. (Similarly, in the spring-coupled mass system investigated in the preceding section, the external forces exerted on an individual spring act to lengthen it when its extension is positive, and vice versa.)
Consider the motion of a thin section of the rod lying between  and  . If  is the mass density of the rod then the section's mass is  . The stress acting on the left boundary of the section is  . Since stress is force per unit area, the force acting on the left boundary is . This force is directed in the minus  -direction, assuming that the strain is positive (i.e., the force acts to lengthen the section). Likewise, the force acting on the right boundary of the section is  , and is directed in the positive  -direction, assuming that the strain is positive (i.e., the force again acts to lengt
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