(II) In deriving Eq. 2, v=√(FT / μ,) for the speed of a transverse wave on a string, it was as

step 1 that for this equation, for this wave, the assumption that the amplitude is much smaller than th

(II) In deriving Eq. 2, v=√(FT / μ,) for the speed of a...

(II) In deriving Eq. $2, \quad v=\sqrt{F_{\mathrm{T}} / \mu,}$ for the speed of a transverse wave on a string, it was assumed that the wave's amplitude $A$ is much less than its wavelength $\lambda$ . Assuming a sinusoidal wave shape $D=A \sin (k x-\omega t),$ show via the partial derivative $v^{\prime}=\partial D / \partial t \quad$ that the assumption $A \ll \lambda$ implies that the maximum transverse speed $v_{\text { max }}^{\prime}$ the string itself is much less than the wave velocity. If $A=\lambda / 100$ determine the ratio $v_{\max }^{\prime} / v.$ $v=\sqrt{\frac{F_{\mathrm{T}}}{\mu}} \quad \quad \left[ \begin{array}{l}{\text { transverse }} \\ {\text { wave on a cord }}\end{array}\right]$ (2)

(II) In deriving Eq. $2, \quad v=\sqrt{F_{\mathrm{T}} / \mu,}$ for the speed of a transverse wave on a string, it was assumed that the wave's amplitude $A$ is much less than its wavelength $\lambda$ .
Assuming a sinusoidal wave shape $D=A \sin (k x-\omega t),$ show via the partial derivative $v^{\prime}=\partial D / \partial t \quad$ that the assumption $A \ll \lambda$ implies that the maximum transverse speed $v_{\text { max }}^{\prime}$ the string itself is much less than the wave
velocity. If $A=\lambda / 100$ determine the ratio $v_{\max }^{\prime} / v.$
$v=\sqrt{\frac{F_{\mathrm{T}}}{\mu}} \quad \quad \left[ \begin{array}{l}{\text { transverse }} \\ {\text { wave on a cord }}\end{array}\right]$ (2)

Key Concepts

Transverse Waves on a String

This concept involves waves that travel along a string or rope with their displacement occurring perpendicular to the wave’s direction of travel. The fundamental properties, such as tension and mass per unit length, determine the wave's speed according to the equation v = ?(F?/?). Understanding this concept is crucial for analyzing how disturbances propagate through the medium.

Sinusoidal Wave Function

A sinusoidal wave function, typically expressed as D = A sin(kx - ?t), is used to represent periodic wave motion mathematically. It encapsulates the essential features of wave phenomena, including amplitude (A), wavenumber (k), and angular frequency (?). This function forms the basis for further analysis, such as determining velocity variations and phase relations along the wave.

Small Amplitude Approximation

The small amplitude approximation, where the amplitude A of the wave is much smaller than its wavelength ?, simplifies the analysis of wave motion by allowing linear approximations to be used. This assumption means that nonlinear effects are negligible, which in turn implies that the transverse motion of individual points on the string is minor compared to the overall propagation speed of the wave.

Wave Speed versus Transverse Particle Speed

This concept distinguishes between the propagation speed of the wave (v) and the instantaneous transverse speed of particles in the medium (v'). While v is determined by the medium's tension and mass per unit length, v' is calculated using the time derivative of the displacement (?D/?t). The small amplitude condition implies that the maximum transverse particle speed is much lower than the speed of the wave itself.

Partial Derivatives in Wave Motion

Using partial derivatives, especially the temporal derivative ?D/?t, is a mathematical method to determine the instantaneous velocity of a point on a wave. This derivative shows how the displacement changes with time, leading to insights into the dynamic behavior of the medium under wave propagation. It is essential for linking the mathematical representation of the wave with its physical characteristics, such as the maximum transverse speed of the string.

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