13 WAVE MOTION 13.2 Waves on a stretched string
tan (13.2)'tan (13.3)Figure 108: Forces acting on a segment of a stretched string.assumed to be infinitesimally small, which implies that the string is everywhere
almost parallel with the x-axis (the string displacement is greatly exaggerated in
Fig. 108 , for the sake of clarity).
Consider the y-component of the string segment’s equation of motion. The netforce acting on the segment in the y-direction takes the form
fy(x, t) = T sin δθ 2 − T sin δθ 1 ' T (δθ 2 − δθ 1 ), (13.1)since sin θ ' θ when θ is small. Now, from calculus,
∂y(x − δx/2, t)
= δθ 1 ' δθ 1 ,
∂y(x + δx/2, t)
= δθ 2 ' δθ 2 ,since the gradient, dy(x)/dx, of the curve y(x) is equal to the tangent of the angle
subtended by this curve with the x-axis. Note that tan θ θ when θ is small. The
quantity ∂y(x, t)/∂x refers to the derivative of y(x, t) with respect to x, keeping
t constant—such a derivative is known as a partial derivative. Equations (13.1)–
(13.3) can be combined to give
∂y(x + δx/2, t) ∂y(x^ −^ δx/2,^ t)^ ∂^2 y(x, t)
fy(x, t) = T
∂x−
∂x(^) = T δx (^) ∂x 2.^ (13.4)^
T
2
1
T x^ −^ x/2^ x^ +^ ^ x/2^ x^ −>^
y^
−>
∂x
∂x