Verified answer

The fixed ends will always be nodes.

To produce wave of largest wavelength,there will be only one antinode in between the two nodes at the ends

distance between adjescent nodes = Half of largest wavelength

distance between adjescent nodes =length of string (L)

Half of largest wavelength = L

largest wavelength = 2L

wavelength = 4m

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in next mode of vibration, there will be 1 more node in between the nodes at the two ends

one wavelength = L

wavelength =2m

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in next higher mode of vibration, there will be 2 more node in between the nodes at the two ends

(3/2) wavelength=L

wavelength = 2L/3

wavelength =4/3 m

wavelengths are 4m, 2m and 4/3 m

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General formula

wavelength =2L/n where n= ( number of nodes - 1 )

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WAVELENGTH larger than 4m say 5 m cannot be produced by this length of string

largest wavelength=2L=twice the length of the string

To answer this you need to draw!

Draw your straight string.

Now draw an arc from one end of the string to the other. The highest point of the arc should be at the middle of the string. Note that this arc is actually only 1/2 a wavelength! What is the wavelength of this wave?

Now draw your string again - straight. This time put a dot in the center. Now draw an arc above the line from the left side of the string, ending at the center dot. Now draw an arc BELOW the line from the center of the string ending on the right side of the string. What is the wavelength of this wave?

finally draw you straight string again and put dots at 1/3 and 2/3 points. Draw arc above, then below, then above. Note this wave has a wavelength LESS than 2m. What is the wavelength.

these are the first 3 modes that will produce standing waves, ANY wavelength other than these will not produce a standing wave....

Let L = the legnth of the string. In this example L=2 meters. The easiest solutions will be the first, second and third harmonics. The first will have a wavelegnth of L, second will be L/2 and third will be L/3. L over any whole number will produce that number of standing waves. If you use any other numbers, you will not get stable standing waves.

simple. A wave length of 1.0 m will give two standing waves with a node in the middle. A wave length of 0.5 m will give 4 standing waves. A wave length of 0.25 m will give 8 standing waves. All these give standing wave lengths because they are multiples of a single standing of 2.0 m and are all harmonics of this wave length. One length that wouldn't produce a standing wave is a wave length that isn't a multiple of one wave length, such as a wave length of 1.2 m long.