The time variation in the water-vapour bands in oxygen-rich Mira variables has been investigated using multi-epoch ISO/SWS spectra of four Mira variables in the 2.5–4.0 \mu m region .
All four stars show H _ { 2 } O bands in absorption around minimum in the visual light curve .
At maximum , H _ { 2 } O emission features appear in the \sim 3.5–4.0 \mu m region , while the features at shorter wavelengths remain in absorption .
These H _ { 2 } O bands in the 2.5–4.0 \mu m region originate from the extended atmosphere .

The analysis has been carried out with a disk shape , slab geometry model .
The observed H _ { 2 } O bands are reproduced by two layers ; a ‘ hot ’ layer with an excitation temperature of 2000 K and a ‘ cool ’ layer with an excitation temperature of 1000–1400 K. The column densities of the ‘ hot ’ layer are 6 \times 10 ^ { 20 } – 3 \times 10 ^ { 22 } cm ^ { -2 } , and exceed 3 \times 10 ^ { 21 } cm ^ { -2 } when the features are observed in emission .
The radii of the ‘ hot ’ layer ( R _ { \mathrm { hot } } ) are \sim 1 R _ { * } at visual minimum and 2 R _ { * } at maximum , where R _ { * } is a radius of background source of the model , in practical , the radius of a 3000 K black body .
The ‘ cool ’ layer has the column density ( N _ { \mathrm { cool } } ) of 7 \times 10 ^ { 20 } – 5 \times 10 ^ { 22 } cm ^ { -2 } , and is located at 2.5–4.0 R _ { * } . N _ { \mathrm { cool } } depends on the object rather than the variability phase .

The time variation of \mbox { $R _ { \mathrm { hot } } $ } / R _ { * } from 1 to 2 is attributed to the actual variation in the radius of the H _ { 2 } O layer , since the variation in R _ { \mathrm { hot } } far exceeds the variation in the ‘ continuum ’ stellar radius .
A high H _ { 2 } O density shell occurs near the surface of the star around minimum , and moves out with the stellar pulsation .
This shell gradually fades away after maximum , and a new high H _ { 2 } O density shell is formed in the inner region again at the next minimum .
Due to large optical depth of H _ { 2 } O , the near-infrared variability is dominated by the H _ { 2 } O layer , and the L ’ -band flux correlates with the area of the H _ { 2 } O shell .
The infrared molecular bands trace the structure of the extended atmosphere and impose appreciable effects on near-infrared light curve of Mira variables .