Two travelling waves produce a standing wave represented by equation `y = 1.0 mm cos(1.57 cm^-1 x) sin(78.5 s^-1` the node closest to origin in the region x > 0 will be at x = ......cm

Answer

To find the position of the node closest to the origin in the region \( x > 0 \) for the standing wave represented by the equation

\[ y = 1.0 \, \text{mm} \cos(1.57 \, \text{cm}^{-1} \, x) \sin(78.5 \, \text{s}^{-1} \, t), \]

we need to follow these steps:

### Step 1: Understand the standing wave equation
The standing wave equation is given in the form \( y = A \cos(kx) \sin(\omega t) \), where:
- \( A \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency.

From the equation, we identify:
- \( A = 1.0 \, \text{mm} \),
- \( k = 1.57 \, \text{cm}^{-1} \),
- \( \omega = 78.5 \, \text{s}^{-1} \).

### Step 2: Identify the condition for nodes
Nodes occur when the displacement \( y \) is zero. This happens when \( \cos(kx) = 0 \) because the sine function can take any value.

### Step 3: Set up the equation for nodes
To find the nodes, we set:

\[ \cos(kx) = 0. \]

The cosine function is zero at:

\[ kx = \frac{\pi}{2} + n\pi, \]

where \( n \) is any integer.

### Step 4: Solve for \( x \)
Substituting \( k = 1.57 \, \text{cm}^{-1} \):

\[ 1.57 \, x = \frac{\pi}{2} + n\pi. \]

For the first node closest to the origin (where \( n = 0 \)):

\[ 1.57 \, x = \frac{\pi}{2}. \]

Now, substituting \( \pi \approx 3.14 \):

\[ 1.57 \, x = \frac{3.14}{2} = 1.57. \]

### Step 5: Calculate \( x \)
Dividing both sides by \( 1.57 \):

\[ x = \frac{1.57}{1.57} = 1 \, \text{cm}. \]

### Conclusion
Thus, the node closest to the origin in the region \( x > 0 \) is at:

\[ \boxed{1 \, \text{cm}}. \]