The question aims to find the **time** it takes for a **wave** generated in a **string tied** to a **wall** to have a **standing wave.**

The question depends on the concepts of **waves** generated in a **string** tied to a **stationary object.** A **standing wave** is generated when two waves with the **same amplitude** and **wavelength** have **interference** and move in **opposite directions.** A **rope** tied to a wall or a stationary rigid object will generate **standing waves.**

The **waves** generated in a **string** are called **transverse waves. Transverse waves** have the wave direction **perpendicular** to the **oscillations** of the **string/rope.** The **velocity** or **speed** of the **wave oscillating** in a **string** is given as:

\[ v = \lambda f \]

Also, **frequency** is given as:

\[ f = \dfrac{ 1 }{ T } \]

It also depends on the **equation** of **motion** as we need to calculate the **time** it takes a **standing** wave to fill the entire **length** of the **cord.** The equation for **time** is given as:

\[ t = \dfrac{ s }{ v } \]

## Expert Answer

The given information about the problem is given as follows:

\[ Frequency\ of\ the\ Wave\ f = 5\ Hz \]

\[ Length\ of\ the\ String\ L = 3.5\ m \]

\[ Wavelength\ \lambda = 1\ m \]

The **velocity** of the **wave** in the **string** can be calculated by the formula, which is given as:

\[ v = f \lambda \]

Substituting the values, we get:

\[ v = 5 \times 1 \]

\[ v = 5\ m/s \]

The **time** that the wave will take to **reach** from one end to the other end is given by the **equation** of **motion** as:

\[ t’ = \dfrac{ L }{ v } \]

\[ t’ = \dfrac{ 3.5 }{ 5 } \]

\[ t’ = 0.7\ s \]

The **total time** taken by the **standing wave** to fill the entire length of the **cord** is given as:

\[ t = 2 \times t’ \]

\[ t = 2 \times 0.7 \]

\[ t = 1.4\ s \]

## Numerical Result

The **total time** taken by the **standing wave** to fill the **entire length** of the **cord** is calculated to be:

\[ t = 1.4\ s \]

## Example

A **rope** is tied to a **steel block** and is shaken from the other end. The **length** of the **rope** is **10m,** and the **wavelength** of the wave generated is **1.5m.** The **frequency** of the waves generated is **10 Hz.** Find the **time** taken by the **wave** to reach from hand to the steel block.

The information given in the problem is as follows:

\[ Frequency\ of\ the\ Wave\ f = 10\ Hz \]

\[ Length\ of\ the\ String\ L = 10\ m \]

\[ Wavelength\ \lambda = 1.5\ m \]

The **velocity** of the **wave** in the **string** can be calculated by the formula, which is given as:

\[ v = f \lambda \]

Substituting the values, we get:

\[ v = 10 \times 1.5 \]

\[ v = 15\ m/s \]

The **time** that the **wave** will take to reach from one end to the other end is given by the **equation** of **motion** as:

\[ t = \dfrac{ L }{ v } \]

\[ t = \dfrac{ 10 }{ 15 } \]

\[ t = 0.67\ s \]